Not to answer Baruch Hu Uvaruch Shemo: The listeners are not to say Baruch Hu Uvarach Shemo upon hearing Hashem’s name said in the blessing. In the event one said Baruch Hu Uvaruch Shemo it is possible that one has not fulfilled his obligation of the blessing. For this reason it is imperative to warn the public of this matter, as many are unaware of this.
Answering Amen: The congregation answers Amen after each of the two blessings that are recited. However some say that those which follow the Sefaradic custom are not to answer Amen after the blessing of Shehechiyanu which is said by the second day. See Q&A above! If one listened to the blessing but did not answer Amen he nevertheless fulfills his obligation.
What is one to do if he is unable to hear the blessing properly and missed a few words?
He is to recite the blessings to himself making sure to conclude the blessing prior to the start of the blows.
If one arrived to Shul after the blessing over Shofar had been recited, may he or she say the blessing on their own?
If one is able to say the blessing prior to the first blow then he should do so. If one is unable to then he [or she] should say the blessing in-between the sets of Tekios. See next Q&A!
If one missed the blessing and heard some of the sounds may a blessing still be recited over the remaining sounds?
Did not yet hear 30 blows: If one did not yet hear 30 sounds then a blessing may be recited over the remaining sounds according to all. Thus if one walked into Shul in middle and he missed even one blow, he can say the blessing prior to hearing the single blow that he missed.
One already heard 30 blows [all Tekios Meyushav] may a blessing be said over Tekios Meumad: If one already heard 30 blows some Poskim rule one may still recite a blessing over the Musaf blows. Others however rule he can no longer recite a blessing and is rather to think it in his mind.
What is one to do if he arrived to Shul before Musaf?
He is to say a blessing to himself prior to the blows of Musaf. If in this congregation the custom is to blow during the silent Musaf prayer then he should explicitly have in mind not to fulfill his obligation with these blows and delay the blessing until the blows in the Chazan’s repetition.
Blowing Shofar for the sake of others if one is not currently fulfilling his obligation:
A person is able to fulfill his obligation of Shofar through hearing the blows from another person even if he knows how to blow Shofar himself and even if there isn’t a Minyan present.
Saying the blessing when blowing for other men: If one who has already fulfilled his obligation of hearing Shofar is blowing Shofar for a man who has not yet fulfilled his obligation, then if the man knows how to recite the blessings, he is to recite them rather than hear the blessings from the blower who already fulfilled his obligation. [This law likewise applies if the blower did not yet fulfill his obligation, but plans to only do so with a later blowing. Bedieved if the blower recited the blessings rather than the listener, nevertheless the listener fulfills his obligation of reciting the blessings.] If however the listener is unable to recite the blessings then the blower is to recite both blessings [i.e. Leshmoa and Shehechiyanu] for the listener. [On the second day of Rosh Hashanah men of Sephardic origin do not recite the blessing of Shehechiyanu.]
Saying the blessing when blowing for only Women: If one already fulfilled his obligation of hearing Shofar, or plans to do so with a later blowing, then if he is blowing for only women, [if the women follow the Ashkenazi custom], she is to say the blessings herself. If she cannot say the blessing herself one may not say the blessings for her and one is hence to blow without a blessing. [If she follows the Sefaradi custom she is not to say the blessing at all even if she knows the blessing.] See Halacha 1B for further details on this topic!
Having intent to fulfill the Mitzvah: It is important to remind those hearing Shofar to have in mind while hearing it, that they are fulfilling the Mitzvah of Shofar, as Mitzvahs require intent. See Halacha 3!
Silence throughout the Shofar blowing:
During the actual blowing of the Shofar, whether of Meyushav [i.e. before Musaf] or Meumad [i.e. during Musaf], it is forbidden for one to [make any noise even to] spit, as one must hear the entire sound from beginning to end, even if the blow is very long. [It goes without saying that one may not cough, burp or make any other noise with his mouth during the blows as this causes others to become disturbed. If one feels the need to spit then he should let it get absorbed in his clothing.]
If there was noise during the Shofar blowing does one nevertheless fulfill his Mitzvah? Even if there were trumpets blowing during the Shofar blowing nevertheless [if one concentrated on the Shofar sounds] he fulfills his obligation. This applies by all forms of noise.
Children: Due to the above [necessity for quiet during the blowing] children which are below the age of Chinuch are not to be brought to the [men’s side of] Shul for Shofar blowing in order so they do not disturb the listeners. However the women may have the children by their side as in any event they are not obligated in the Mitzvah.
One must hear the entire blow: One is obligated to hear the entire blow even if it is very long.
Must the listeners focus on the sounds during the blowing? Does one fulfill his obligation even if he was thinking of other matters?
One must focus his attention on the sounds during the blows. If one did not do so it is questionable whether he fulfills his obligation of that specific blow in which his mind was drifting. Thus it is proper for one to look inside the Machzor and follow along by each sound. One is to think by each sound how he is fulfilling the will of G-d and giving Him pleasure above. Doing so will help him concentrate and prevent him from allowing his mind to drift. Accordingly one is to beware not to think too deeply while hearing the sounds even of matters that pertain to the Shofar, such as its laws and Kabalistic intents.
If one spoke during the blowing has he fulfilled his obligation?
Seemingly if one spoke during the blowing he does not fulfill his obligation for those blows that were sounded while he was speaking.
 124/2; Degul Merivava, Chayeh Adam and other Achronim brought in M”B 124/21; See Shraga Hameir 5/55; Piskeiy Teshuvos 585 footnote 29; 124/11 footnote 106; Mishneh Halachos 13/16
 The reason: Whenever a person is hearing a blessing with intent to fulfill his obligation, such as by Kiddush, Havdala and blessings over fruits and Mitzvos, he may not answer Baruch Hu Uvaruch Shemo, being that one who hears [and fulfills his obligation] is like one who is saying it himself and it is forbidden to make an interval in the midst of a blessing. [Admur 124/2 in parentheses]
 Admur ibid; Chayeh Adam brought in M”B 124/21; See Shraga Hameir ibid that in such a case he also missed the words “Elokeinu Melech Haolam of the person saying the blessing; See Minchas Elazar 1/20
The reason: It is possible that even Bedieved one does not fulfill his obligation as since he made an interval between Hashem’s name and Malchus with words that were not inserted by the Sages he has therefore swerved from the dialect of the blessings that was set by the Sages. [Admur ibid in parentheses]
Other opinions: Some Poskim rule that if one answered Baruch Hu Uvarach Shemo he nevertheless fulfills his obligation. [conclusion of M”B ibid; Mahram Shick 51; Birkeiy Yosef 213/3 in name of Mahrash Abuhav; Piskeiy Teshuvos 124/11 and 585/3]
 Admur 213/4; Michaber 213/2
 Ben Ish Chaiy Netzavim 15; Kaf Hachaim 585/13; Piskeiy Teshuvos 585/3
 Kaf Hachaim 585/39; Piskeiy Teshuvos 585/8
 Kaf Hachaim 585/37 and 39
 Kaf Hachaim 585/36; Peri Chadash
 Peri Haretz 9; Ikarei Daat 30/1; Bnei Yissachar 3/34 [even after Musaf if there remain the ten blows he may say the blessing]
 Kaf Hachaim 585/36 and 39
 Piskeiy Teshuvos 585/8 footnote 59
 591/1; Michaber 591/1; M”B 594/1
The reason: This follows the same law as by all Mitzvos that one person is able to be Motzi another. [Tur] This is due to the concept of Areivus that every Jew is a guarantor for each other. [Kaf Hachaim 594/1]
 585/5; M”A 585/3 based on Terumas Hadeshen 140; Opinion in M”B 585/5 and 692/10; Machatzis Hashekel 585; P”M 585 A”A 3; Derech Hachaim 2 regarding Shofar; Mateh Efraim 585/6; Chayeh Adam 141/7; Kaf Hachaim 585/21; Poskim who rule similarly in other Halachas:
Kiddush: Michaber 273/4; Admur 273/6; Megillah: Kaf Hachaim 692/14 and 25; Derech HaChaim 3; Minchas Yitzchak 3/53-54; Salmas Chaim 269; Luach Tukichinsky; Kinyan Torah 3/103; See Sdei Chemed R”R 2/19
The Rama [585/2] rules that the blower is to recite both blessings whether he is blowing for himself, to fulfill his personal obligation, or whether he is blowing for others. The M”A [585/3] brings the Terumas Hadeshen  which rules that it is always better for the listener to recite the blessing, although he concludes that the custom is that the blower recites it. Regarding Shehechiyanu the Terumas Hadeshen does not discuss who is to recite it. The Rambam, Mateh Moshe, Mahriv and Hagahos Maimanis rule that the listener is to recite it while the Beis Yosef writes that the custom is for the blower to recite it. It is based on these two customs written in the Terumas Hadeshen and Beis Yosef that the Rama here writes that the blower is to recite both blessings. Nevertheless the M”A ibid concludes that this custom only applies if the listener does not know how to recite the blessing however if the listener knows how to recite it, then he is to do so and he then fulfills his obligation according to all. The ruling of Admur here follows the ruling of the M”A ibid.
Ruling of the M”B: The M”B 585/5 rules that the custom is to be lenient and one is not to protest their opinion as so is the main opinion. The Kaf Hachaim 585/21 questions this ruling of the M”B as majority of Poskim ibid rule that it is to initially be said by the listener. However see Minchas Yitzchak ibid that explains the M”B did not mean to say that most Poskim initially rule this way.
 The reason: This follows the general rule that the blessings are to be recited by one who is currently fulfilling his obligation of the Mitzvah. [273/6] The Terumas Hadeshen 140 explains that since the Mitzvah is to hear the sound of the Shofar and both the blower and listener are equally fulfilling this action of hearing, therefore it is better for the listener to recite the blessing since he is fulfilling his obligation with the hearing. [This is unlike the law of placing a Mezuzah for another person in which case the person placing the Mezuzah says the blessing, as in that case he is the only person doing the action as opposed to the owner of the house.]
 See 273/6; Thus if one is blowing for other Jews prior to Shacharis and he plans to fulfill the Mitzvah during Davening, the listeners are to recite the blessings if they have knowledge in how to do so.
 273/6; Terumos Hadeshen 140
 Michaber 600/3; Admur 600/7
 589/2; Rama 589/7
 Michaber 589/6- as is the ruling of the Beis Yosef in all Mitzvos that women are not obligated in.
Other Opinions-Sefaradim: Some Sefaradi Poskim rule that women may recite a blessing over any Mitzvah that they are not obligated in. [Birkeiy Yosef 654/2; Yosef Ometz 82; See Kaf Hachaim 589/23] The Birkeiy Yosef ibid states that after seeing the answer from heaven in Shut Min hashamayim he retracted his ruling like the Michaber and would tell women to recite a blessing. He concludes that if the Michaber would have seen the Shut Min hashamayim certainly he would have retracted his ruling.
 592/8; M”A 592/2; M”E 592/3; M”B 592/10; Kaf Hachaim 592/16
 M”A 592/2
 As explained in 590/10
Is this prohibition due to fear that one may not hear the sound of the Shofar or due to fear that one may lose concentration, and one is required to concentrate during the hearing of the blows? It is evident from the ruling below regarding children that the issue is not only that one may not hear the sound but that he may lose concentration on the sound. Hence from here it is evident that one must concentrate on hearing the Shofar sound. See Q&A!
 Darkei Moshe 589/3 in name of Mahril; Elya Raba 587/6; M”E 592/3; M”B 592/10; Kaf Hachaim 592/16; 587/19; The Darkei Moshe ibid writes that it was for this reason that the Mahril avoided eating nuts on R”H in order so he does not have to spit during the Tekios.
 M”E 592/3; M”B 592/10; Kaf Hachaim 592/16
 590/14; See Piskeiy Teshuvos 587 footnote 6
 As he has certainly focused on the blow hence tuning the noise out of his attention. As by a Mitzvah which occurs only once a year certainly one pays extra attention to it, as it is beloved on to the person. Hence even if there was a trumpet blowing at the same time [and making similar noise] one fulfills his obligation. [ibid] Vetzaruch Iyun from 587/1 [“One is required to hear the pure sound of the Shofar blow without any other sounds intertwined. If one heard mixed sounds he has not fulfilled his obligation.”] and 592/8 [next line in this Halacha] regarding spitting and children, which implies that if there is any noise during the blow one is not Yotzei. One must hence say that there is a difference between one who concentrated on the sound, in which case he is Yotzei even if he heard another sound, and if he heard noise and did not concentrate, in which case he is not Yotzei. See Q&A! Similarly one must say there is a difference between an echo of the Shofar which sounds like the Shofar sound and cannot be differentiated and other sounds. See Piskeiy Teshuvos 587 footnote 7
 592/8; Drashos Mahril R”H; Kneses Hagedola 589; M”A 589/1; Elya Raba 589/1; P”M 589; M”E 589/4; Kaf Hachaim 587/19; 589/5
 The listeners may become disturbed by watching the children and seeing their mischievous actions. [Mahril ibid]
 See Piskeiy Teshuvos 589/6; 587/3 footnote 8
 So is evident from the ruling above regarding not to bring children below Chinuch to Shul and the prohibition of spitting during Shofar.
 Piskeiy Teshuvos 589/6; 587/3 footnote 8; See the following Poskim regarding Megillah: Levushei Serud on M”A 590/15; see also Nemukei Orach Chaim 690; Piskeiy Teshuvos 690/4; Eretz Tzevi 1/45; Shearim Hametzuyanim Behalacha 141/13; This commonly occurs to Torah scholars and business men that their minds wander to Torah thoughts and business dealings when hearing something read for a long period of time. [Nemukei Orach Chaim ibid]
 Yesod Veshoresh Havoda 11/3
 Piskeiy Teshuvos 589 footnote 28
 See previous Q&A! See 592/8; Piskeiy Teshuvos 587/3; 585 footnote 32