Answer: The Gemara (Berachos 4b; 9b; 42a) writes that one should ensure to make no interruption between the berachos after shema and the amida, especially during shacharis (Rashi Berachos 4b). One who is particular to do so will be protected that day from harm.
The Shulchan Aruch (OC 66:7; 111:1; 236:2) writes that one should not make any hefsek, unnecessary interruption between go’al yisrael and the amida. Therefore, one must not answer amen. However, the Rema quotes the Tur (OC 66) who does not consider it to be a hefsek and writes that one should answer amen to the chazzan’s beracha. The Aruch Hashulchan (OC 66:14) writes that the minhag is to follow the Shulchan Aruch and avoid saying amen.
In order to avoid this safek, the Aruch Hashulchan (OC 66:15) and Mishna Berura (66:35) write that one should aim to finish the beracha of go’al yisrael together with the chazzan, thereby exempting oneself from responding at all.
While R’ Ephraim Greenblatt (Rivevos Ephraim 1:71; 6:42:1) explains the rationale behind the practice for many chazzanim to end this beracha quietly, he quotes R’ Yosef Eliyahu Henkin (Edus L’yisrael, Beis Hakenesses 1:64) who often spoke out against this practice. Thus, in his Ezras Torah calendar, it is written that according to R’ Henkin, ‘a chazzan who says the ending of go’al yisrael in an inaudible voice, is violating the Talmud’s ruling. Therefore, it is a mitzvah to correct this matter, and to insist that the chazzan begin yotzer ohr and conclude go’al yisrael in an audible voice.’
R’ Shlomo Zalman Auerbach (Halichos Shlomo, Tefilla 7:18) notes that throughout the discussion among the poskim as to whether to say amen, they did not propose saying it quietly. Likewise, R’ Moshe Sternbuch (Teshuvos Vehanhagos 1:105) writes that if the chazzan does not end this beracha loudly, he is preventing those at other points in davening from saying amen. He demonstrates that this follows Rambam, too (Tefilla 9:1).
In conclusion, it is important for the chazzan to end the beracha go’al yisrael before amida out loud.