I should have been more careful with what I wrote about this yesterday. The שו"ע in סימן רי"ג brings down this din, that you should not be יוצא על המחיה from someone else. However, the Mishna Berura there comments that nowadays since people are מזלזל in these berachos we can be סומך on one person being מוציא everyone. The Mishna Berura then says that לכתחילה each person should say it word for word.
R' Shachter quotes the Rav in his sefer פניני הרב (page 58), that the Rav was in YU once for Shabbos and saw this being done (one person being מוציא everyone) and was not happy with this and protested.
I remembered the halacha from the Rav and therefore wrote what I wrote yesterday.