Q. Boruch Hu Uvoruch Shemo & Omein: Should they be said aloud or can they be said quietly?

A. Shulchan Aruch (124: 12) quoting Talmud (Brochos 45a) rules that one who answers omein should not rise his voice more than the one reciting the brocho. However, Meforshim question that it seems to contradict the teaching of the Talmud (Shabbos 119b) that if one who answers omein, with all his might the gates of Gan Eden are opened for him. Horav Shlomo Miller Shlit’a explained that this follows the opinion of those who maintain that “bechol kocho” does not mean with a loud, high volume voice, but with his greatest intention. (Ritvo ibid, quotes Rabenu Yona that it actually means loud and clamorous.
Mishna Berura (ibid. 47) maintains that if one’s intention is to rouse and energize others into answering with better and stronger kavana and intent, it is permitted. (Shaarei Brochos p. 358, quotes othe answers and opinions).
Mishna Berura (ibid.) also asserts that the above applies to answering borchu or the zimun blessing before birchas hamozon. Presumably the same would apply to answering baruch Hu, ubaruch Shemo.
Horav Shlomo Miller’s opinion is that baruch Hu, ubaruch shemo does not carry the same import and significance as omein, and may be recited with less intensity. He added that the minhag of the Gr’a, Brisk and others is not to answer baruch Hu, ubaruch shemo at all.

Rabbi A. Bartfeld as revised by Horav Shlomo Miller Shlit”a