newhere wrote I once saw that Rav Shlomo Zalman zt’l held not to say it out loud for the same reason, but i dont remember where, probably somewhere in halichos shlomoh. This would obviously not apply to bentching which is not a silent teffilah. See ?????? ????? ???? ?”? ?????? ???? note #20
from notes of talmidim. Reason cited ??? ????? applies to Bentching as well.
But Debreciner Rav ZT”L in Shut Beer Moshe Vol 4 siman 10 is matir for one only, after that it is Assur