Before the sounding of the Shofar, the Hazan recites the Beracha, "Baruch…Asher Kideshanu Be’misvotav Ve’sivanu Li’shmoa Kol Shofar," and the congregation answers "Amen." The congregation should not answer "Baruch Hu U’baruch Shemo" during the Beracha, because anytime one hears a Beracha in order to fulfill his obligation to recite it, he may not answer "Baruch Hu U’baruch Shemo."
If a person suspects that he will not hear every word of the Beracha, such as if he has a hearing impairment or his seat is far away from the Teba, then he should recite the Beracha by himself as the Hazan recites the Beracha. The Halachic authorities debate the question of whether a person in such a case should answer "Amen" to the Hazan’s Beracha if he finishes his Beracha before the Hazan. For example, if a person sits far away from the Hazan and thus suspects that he might not hear each word of the Beracha, and so he recites the Beracha on his own, and after completing the Beracha he suddenly hears the Hazan recite, "Li’shmoa Kol Shofar," does he answer "Amen"?
Hacham Ovadia Yosef, in his Yabia Omer (8:20:4), rules that one may not answer "Amen" in such a case, as this would constitute a "Hefsek" (unlawful interruption) between the Beracha and the Misva. Once one recites a Beracha over a Misva, he must immediately proceed to perform the Misva, and thus after reciting the Beracha over the Misva of Shofar, one must then sound or hear the Shofar without any interruption. Hacham Ovadia draws an analogy to one who recites Kiddush quietly to himself while the head of the household recites Kiddush (such as if he is afraid he might not hear every word), and after finishing his Beracha he hears the head of the household finish his Beracha. Clearly, one should not answer "Amen" to another Beracha after reciting Kiddush before he drinks his wine. By the same token, one who recites the Beracha over the Shofar may not make any interruption – including answering "Amen" to the Hazan’s Beracha – before fulfilling the Misva by hearing the Shofar sound.
Nevertheless, Hacham Ovadia adds, if one mistakenly answered "Amen" to the Hazan’s Beracha, he has nevertheless fulfilled his requirement and does not have to repeat the Beracha. Since the Hazan’s Beracha also relates to the Misva of Shofar, we may, after the fact, consider the "Amen" response germane to the Misva such that it does not constitute a Hefsek. But from the outset, one should ensure not to recite "Amen" to the Hazan’s Beracha in such a case, and simply wait silently for the sounding of the Shofar.
This ruling is codified in Yalkut Yosef (p. 172; listen to audio recording for precise citation).
Summary: If one suspects that he will not hear every word of the Hazan’s Beracha over the Shofar, he should recite the Beracha by himself and then wait silently for the sounding of the Shofar. He should not answer "Amen" if he hears the conclusion of the Hazan’s Beracha.