The Shofar is sounded during the repetition of the Amida of Musaf on Rosh Hashanah, after the conclusion of the Malchuyot, Zichronot and Shofarot sections. According to some opinions, the Shofar sounds blown during the repetition of the Amida are the primary Shofar sounds, and are more important than even the Shofar sounds blown during the silent Amida.
Since the Sages instituted that the Shofar be blown in the framework of Malchuyot, Zichronot and Shofarot, it is especially important for one to pay attention to the repetition of the Amida. If a person is not paying attention, and is instead involving in other activities, then he is not considered to have heard the Shofar in the framework of these Berachot. The work Ner Le’sion (p. 262; listen to audio recording for precise citation) elaborates on the importance of silently following the Hazan with a Siddur during the repetition of the Amida of Musaf on Rosh Hashanah, and writes that one must not learn or engage in conversation during the repetition, and must instead follow along with the Hazan.
In truth, the Ner Le’sion notes, it is in any event strictly forbidden to speak during the repetition of the Amida at any time, and the Shulhan Aruch (Orah Haim 124:7) and Ben Ish Hai (Parashat Teruma) go so far as to say that if one engages in conversation during the repetition of the Amida, "His sin is too great to bear." The repetition of the Amida is on a higher level than even the silent Amida, and thus just as we would never think to speak during the silent Amida, we should not even consider speaking during the repetition.
Moreover, one must ensure to respond to every Beracha with "Baruch Hu U’baruch Shemo" and "Amen," and to concentrate during these responses. The importance of "Baruch Hu U’baruch Shemo" and "Amen" cannot be overstated. The Pele Yoetz (Rav Eliezer Papo, 1786-1827) cites a comment from the Zohar that one who is not careful about responding "Baruch Hu U’baruch Shemo" and "Amen" is punished in the lowest levels of Gehinam, and of those who respond without intention it is said, "U’bozai Yekalu" ("and those who scorn Me shall be belittled" – Shemuel I 2:30). He adds that if people realized the great reward for answering "Baruch Hu U’baruch Shemo" and "Amen," they would run about searching for opportunities to hear Berachot as though they were frantically searching for treasures.
Summary: While one is always required to listen silently and attentively to the repetition of the Amida, this is especially important when it comes to the repetition of the Amida of Musaf on Rosh Hashanah.