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.
 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