Counting with a blessing after overhearing the Sefira/blessing from another:
If one casually overheard the counting of another person, whether from a private individual or congregation, without any particular intent of being Yotzei or not being Yotzei, then it is disputed as to whether he fulfilled his obligation. This dispute applies even if the person who said the Sefira did not have in mind to fulfill the obligation of the listener. Practically, one who overheard the Sefira from another is to repeat the Sefira without a blessing. This applies beginning from Bein Hashmashos. If however one specifically had in mind to not fulfill his obligation with the overhearing of the Sefira, then he is to repeat the Sefira with a blessing. If however he did not have in mind to not fulfill his obligation, then even if he had in mind that he is not intending to fulfill his obligation, he is to repeat the counting without a blessing, as stated above. [Accordingly, whenever one overhears the blessing/Sefira from another he is to have in mind to not be Yotzei with the blessing/Sefira. It is also proper for the person saying the blessing/Sefira, such as the Chazan, to specifically have in mind to not be Motzi with his blessing and Sefira anyone who plans to say the blessing afterwards. To emphasize this point that one is not being Yotzei with the Chazan, one is to answer Baruch Hu Uvracuh Shemo to his blessing. In a case that one overheard the Sefira, and was supposed to repeat the counting and did not do so, see Halacha 20 that if he had in mind to be Yotzei then he may nevertheless continue counting with a blessing the following nights, while if he did not have anything in mind then it is questionable as to whether he may continue to count with a blessing.]
During a Minyan, must one have intention to not fulfill his obligation with the Chazzan’s counting?
It is customary for the Chazan to recite the blessing and Sefira aloud, which is then followed by the blessing and Sefira of the congregation. If one is Davening with this Minyan and plans to recite the Sefira together with the congregation then it is not necessary to have in mind to not fulfill the Mitzvah/blessing with the Chazan, as even if one did not have this explicitly in mind he may continue and count with a blessing. [If however, he does not plan to count Sefira at this time, or he is still holding in Shema or Shemoneh Esrei, then he must explicitly have this in mind.]
If in middle of a Minyan for Maariv, or while waiting for a Minyan for Maariv, one overheard the counting of Sefira from another nearby Minyan, may he count with a blessing?
One may only count with a blessing if he had in mind to not fulfill his obligation with their counting. If he did not have in mind to not be Yotzei, then he is to recount without a blessing. This applies even if one was in the midst of Shemoneh Esrei, when he heard the Sefira, he nevertheless must have in mind to not be Yotzei. Due to this reason, it is proper for Minyanim to not be within ears distance of each other by Maariv.
If one overheard the counting from a child or woman, may he still count with a blessing?
An adult male [i.e. over Bar Mitzvah] who overheard a woman or child say the blessing/Sefira does not fulfill his obligation according to any opinion and is thus to repeat the Sefira with a blessing.
Having in mind before the start of Sefirah to not be Yotzei with anyone else:
A good advice to avoid entering into the above Halachic issue is to have in mind prior to the start of Sefira that from now and onwards one intends to not be Yotzei with anyone’s blessing/Sefira, unless he decides otherwise [such as if he needs to hear the blessing from another due to Safek]. This intent is effective throughout the entire duration of Sefira and prevents one from needing to have this mind every particular time he overhears a Sefira/blessing.
 Admur 489/12 regarding the Sefira; 213/4 (in parentheses) regarding all blessings [See however Admur 296/17; 6/9, explained below]; Rashba 1/458, brought in Beis Yosef 489, regarding blessings; Rama 489/3 regarding blessing of Sefira “If he had in mind to not be Yotzei”; M”A 489/8, explained in Machatzis Hashekel and P”M 489 A”A 8; Elya Raba 489/12; M”B 489/17 limits this ruling to only blessings [and not verbal Mitzvos]; Michaber 6/4 regarding Birchas Hashachar “They intend to not be Yotzei”; Tehila Ledavid 6/4 in understanding of Michaber ibid [See however Michaber 489/3 and M”B 48918 who implies one is not Yotzei unless he has Kavana to be Yotzei] See Piskeiy Teshuvos 489/8;
Source of Admur’s ruling: This ruling of Admur that one can fulfill a verbal Mitzvah through overhearing another is a novelty that is not explicitly recorded in Poskim prior to Admur. However, it is rooted in a response of the Rashba 1/458 regarding whether one may repeat a blessing after hearing it from the Chazan, to which the Rashba answer’s that one may only do so if he had in mind to not be Yotzei, however if he had no Kavana then he is Yotzei according to those who rule that Mitzvos don’t need Kavana. This ruling of the Rashab is recorded and hinted to in the Michaber ibid and Rama ibid as well as the other Poskim. The novelty of Admur ibid is that he extends this ruling even to the Mitzvah of Sefira. It is possible to learn this way as well in the M”A ibid. as explained in the Machatzis Hasehkel and P”M ibid.
 This dispute is recorded in Admur 489/12 regarding Sefiras HaOmer and 213/4 regarding all blessings “One is to suspect for their words and beware not to enter into a Safek Bracha Levatala.”
Some Poskim [Rashba Brachos 13b; Ran R”H 3 in name of Geonim] rule that Mitzvos do not need intention for one to fulfill his obligation even if they are of Biblical nature. Furthermore, even according to those Poskim [Bahag Brachos 2/7; Rif R”H 3; Rambam Shofar 2/4; Rosh R”H 3] who argue that Biblical Mitzvos require intention, today the Mitzvah of Sefira is Rabbinical, and there are Poskim who rule [Bach 475; M”A 60/3; Admur 60/5] that by a Rabbinical Mitzvah according to all one does not need intention. Hence according to these two groups of Poskim one has fulfilled his obligation upon overhearing the count, as one who hears is considered like one who said it himself. [Admur 489/12] However other Poskim [Peri Chadash 475/4; M”A 489/8; Admur 60/5] rule that even Rabbinical commands retain the same dispute as Biblical commands, and hence in their opinion one who did not have explicit intent does not fulfill his obligation with the Sefira that he overheard.
 Admur 489/12; 213/4; Implication of Rashba ibid
The reason: As one who hears is like one who said it. [Admur ibid] Accordingly, one is to be stringent like both opinions and hence is to repeat the Sefira [to suspect for the stringent opinion who rules that one is not Yotzei without intent] but without the blessing [in order to suspect for those opinions that rule one is Yotzei without intent].
Why is the blessing not recited due to Sfek Sfeka? Why is a blessing not allowed due to this case being a Sfek Sfeka of a blessing, as ruled in 489/24: The Sfek Sfeka is as follows: 1) Perhaps one must always count himself, as rules some Poskim in 489/1. 2) Even if one may hear someone else count, perhaps he must have intention to fulfill his obligation? Furthermore, in the case of Admur 489/12 a third doubt added, as it refers to one who heard the count by Bein Hashmashos, and hence perhaps it was not even night and he has not fulfilled his obligation according to all! Vetzaruch Iyun! Perhaps however one can answer that there is a difference between the forms of Sfek Sfekas in this case and in the case in 489/24 [as brought in Yoreh Deah 110-111, and ruled in Admur 438/9] that a Sfek Sfeka is only given value when the two doubts each come with a separate claim, and thus there are two different claims for why one may continue counting Sefira. When however, both doubts are making the same claim, that one was already Yotzei the Sefira, then we consider it all one doubt, and rule Safek Brachos Lihakel. Thus in 489/12 that all the doubts create the single claim of “he was already Yotzei”, it does not help to remove the Safek Brachos Lihakel rule from implementation. However, in 489/24 in which the doubts are two different claims [perhaps yesterday’s count was valid and even if it isn’t perhaps today’s count is valid] we implement the rule of Sfek Sfeka which overrides the Safek Bracha Lihakel.
Why is one required to repeat the count at all? Tzaruch Iyun why at all one must repeat the count if we rule that Safek Derabanan Lekula, and hence since there are Poskim that rule one fulfilled his obligation, why is the repetition required. Perhaps one can say that only in the case of Admur did he require the repetition to be made, as the case of Admur 489/12 deals with one who said or heard the Sefira by Bein Hashmashos, in which case there is a Sfek Sfeka as whether one is Yotzei or not. If, however one heard it by night, then perhaps Admur would rule that he does not even have to repeat the Sefira. This especially applies if one verbally said the Sefira at night and simply did not have in mind to be Yotzei, in which case there is only one doubt and hence one should not be required to repeat it. [see 489/14 that Admur rules one must repeat] Vetzaruch Iyun!
Contradiction from Admur 6/9: In 6/9 Admur rules that one may repeat Birchas Hashachar if he did not have in mind to fulfill his obligation. This is a change in wording from the Michaber 6/4 who says “They intend to not be Yotzei”. The wording of the Michaber ibid is based on the Rashba, while that of Admur is based on the Iggur. Vetzaruch Iyun Gadol, as this seemingly contradicts the above ruling from 489/12 and 213/4 which does not allow one to repeat the blessing unless he had in mind to not be Yotzei as writes Michaber ibid. [Tehila Ledavid 6/4 asks this question and concludes with a Tzaruch Iyun Gadol] Some suggest the following explanation: By Birchas Hashevach, which is the case in 6/9, Admur holds one must have in mind to be Yotzei from another in order to fulfill his obligation, as otherwise it cannot be considered that he praised anyone. However, by Birchas Hanehenin or Mitzvos, according to one opinion even if he does not have in mind to fulfill his obligation he is nevertheless Yotzei. [Chikreiy Halachos 9/8]
Contradiction from Admur 296/17: In the end of 296/17 it is implied that if one does not have in mind to fulfill Havdala then he must repeat Havdala later. This seemingly contradicts the ruling that one possibly fulfills his obligation even if he simply heard a blessing without having in mind, and it is thus only questionable if he must repeat Havdala. See Tehila Ledavid 296/4 who raises this question. However seemingly the meaning here of Admur is as he wrote earlier in that same Halacha that “if one had in mind to not fulfill his obligation” and hence here too in the conclusion of the Halacha Admur’s intent is not that one did not have in mind to be Yotzei but rather that he had in mind to not be Yotzei.
 The case in Admur 489/12 refers to Bein Hashmashos; See Admur 489/12 who rules that if one counted by Plag Hamincha he is not to repeat the count later on with a blessing. However, in 489/15 Admur rules that if one accidently answered the count by Plag Hamincha, he is to count that night with a blessing, hence ruling that to forfeit the blessing by Plag Hamincha, one must have in mind to fulfill the Mitzvah.
 489/12 regarding Sefiras Haomer and so applies to all Brachos; Rama 489/3; M”A 489/8; Rashba ibid, brought in Beis Yosef 489; Michaber 6/4 “They intend to not be Yotzei”; Tehila Ledavid 6/4 in understanding of Michaber ibid
The reason this intent helps even according to those that rule the Mitzvos do not need intent: As even according to the opinion who holds that that Mitzvos do not need intent to fulfill ones obligation, if one specifically has in mind to not fulfill his obligation everyone agrees he is not Yotzei against his will. [489/12; Rama 489/3; M”A 489/8; Chok Yaakov 489/14] However it is not enough to simply have in mind to not have intention to fulfill his obligation with him; rather he must have in mind to not fulfill his obligation. [489/12]
 Meaning that he negated intent to be Yotzei but did not have intent to not be Yotzei.
 Admur 489/12
The reason: As having in mind to not be Yotzei does not help at all according to those who hold that one does not need to have in mind at all in order to be Yotzei. [Admur ibid]
 Yeshuos Moshe 3/49; Piskeiy Teshuvos ibid
 Yeshuos Moshe ibid; Piskeiy Teshuvos ibid
 Yeshuos Moshe ibid; Piskeiy Teshuvos ibid
 Divreiy Nechmia 489/12; Yeshuos Moshe ibid; Piskeiy Teshuvos ibid and footnote 39
 The reason: Although based on Admur 489/12 one must have explicit intention to not fulfill his obligation, nevertheless, in this case it is not necessary, as the custom is for everyone to count immediately after the Chazan and is hence exactly similar to the ruling in 489/14 regarding one who asks someone else what day it is, in which case we say that he is not Yotzei being that it is considered that the person had explicit intention to not fulfill his obligation with that person. The following is the ruling there which is exactly similar to the question at hand “The reason for this is because at the time of the hearing of this reply the questioner had intent to repeat the counting for himself to fulfill the Mitzvah, as that is precisely the motivation behind his query and subsequent hearing of the answer. Thus, in conclusion, it is considered that he explicitly had in mind to not fulfill his obligation with the hearing of the counting of the answerer. This is in contrast to the person who was asked and replied as he did not have explicit intent at the time of his reply that he desires to repeat the count later on afterwards. Thus, although in truth subconsciously the answerer did want to repeat the count, nevertheless since he did not have this explicitly on his conscious mind at that time, he has therefore fulfilled his obligation according to those who rule that Mitzvos don’t need intent.” [Admur ibid]
 Piskeiy Teshuvos ibid
 As one can only be Yotzei with another “If he heard from one who is obligated in this Mitzvah” [Admur 489/1] and a woman is not obligated in Sefiras Haomer. [Admur 489/2] Likewise, male children may be under debate if they are even obligated at all in this Mitzvah [See Halacha 2C!] and even if they are, their obligation is merely due to Chinuch. [See Michaber 689/2 based on Tana Kama in Megillah 19b and so rules: Rosh; Bahag; Levush; Bach; Olas Shabbos 689/3; Peri Chadash; Beis Yosef 689 in name of Levush; Peri Megadim 689 A”A 1; M”B 689/6; Kaf Hachaim 689/11]
 Piskeiy Teshuvos 489/8 based on Michaber Y.D. 274/1 regarding Leshma of Sefer Torah that it can extend for the entire duration of the writing [aside for the Azkaros]; See also Chelkas Yehoshua 4/4; Sefer Sefiras Haomer end of chapter 8