If a person is reciting the silent Amida along with the Hazan’s repetition of the Amida, he should remain silent during the recitation of Birkat Kohanim. After he and the Hazan recite the Beracha of "Ha’tob Shimcha U’l’cha Na’eh Le’hodot," he should pause and listen to Birkat Kohanim. In such a case, he answers "Amen" to each of the three Berachot of the Kohanim; he answers "Amen" after "Ve’yishmerecha," "Viyhuneka," and "Shalom." Although generally one does not interrupt his recitation of the Amida to answer "Amen" to Birkat Kohanim, a person who is reciting the Amida along with the Hazan’s repetition indeed answers "Amen."
Even in such a case, however, the individual does not answer "Baruch Hu U’baruch Shemo" during Birkat Kohanim.
There is a debate among the Halachic authorities as to whether a person in this case answers "Amen" to the introductory Beracha recited by the Kohanim ("…Asher Kideshanu Bi’kdushato Shel Aharon Ve’sivanu Le’barech Et Amo Yisrael Be’ahaba"). Hacham Ben Sion Abba Shaul (Israel, 1923-1998) ruled that one should not answer "Amen" to the introductory Beracha in this case, whereas Hacham Ovadia Yosef, in his work Yabia Omer (vol. 8), writes that one answers "Amen." Hacham Ovadia brings several proofs to his ruling. As for the final Halacha, it would seem that one may rely on Hacham Ovadia’s ruling and answer "Amen" even to the introductory Beracha. It thus emerges that a person who recites the Amida along with the Hazan’s repetition would make four "Amen" responses – to the Kohanim’s introductory Beracha, and to each of the three verses of Birkat Kohanim.
It is customary for the Kohanim to quietly recite a special "Yehi Rason" prayer before Birkat Kohanim, immediately following "Modim De’rabbanan." The Kohanim should recite this prayer slowly, such that they conclude the prayer just as the Hazan concludes his Beracha with "Ha’tob Shimcha U’l’cha Na’eh Le’hodot." This way, when the congregation answers "Amen" to the Hazan’s Beracha, their response will also refer to the Kohanim’s "Yehi Rason" prayer. The congregation should be aware of this, as well, and realize that their "Amen" response refers both to the Hazan’s Beracha and to the prayer recited by the Kohanim.
It is also customary for the Kohanim to recite a "Ribon Ha’olamim" prayer immediately following Birkat Kohanim. This prayer, too, should be recited slowly so that the conclusion coincides with the Hazan’s recitation of the concluding Beracha "Ha’mebarech Et Amo Yisrael Ba’shalom." Once again, the congregation’s response of "Amen" refers both to the Hazan’s Beracha and to the Kohanim’s prayer. On Rosh Hashanah and Yom Kippur, when several special passages are recited after Birkat Kohanim, in addition to the standard text of "Sim Shalom," the Kohanim should not begin reciting the "Ribon Ha’olamim" prayer until the Hazan is in the middle of "Sim Shalom." Otherwise, they quite obviously would be unable to recite the prayer slowly enough to conclude together with the Hazan’s conclusion of the Beracha. This is the ruling of the Hesed La’alafim (Rav Eliezer Papo, 1770-1828).
Summary: If a person is reciting the silent Amida along with the Hazan’s repetition of the Amida, he listens silently to Birkat Kohanim, answering "Amen" to the Kohanim’s introductory Beracha and to each of the three verses of Birkat Kohanim. He does not answer "Baruch Hu U’baruch Shemo." The Kohanim customarily recite special prayers before and after Birkat Kohanim. The prayer before Birkat Kohanim is recited immediately following "Modim De’rabbanan," and the Kohanim should conclude the prayer just as the Hazan recites "Ha’tob Shimcha U’l’cha Na’eh Le’hodot." Similarly, they should conclude the prayer after Birkat Kohanim just as the Hazan concludes the Beracha with "Ha’mebarech Et Amo Yisrael Ba’shalom."
