I learned that if two groups were eating in two different rooms but even one person could see someone in the other room, then they could have one zimun (I thought this was in Masechet Brachot, but I can't find it). Could this extend to video-chat?
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Voice over the internet is halachically considered an echo. Brachot and prayer can not be said over the internet, nor can you listen to a shofar blast over the phone etc. (source http://www.chaburas.org/ramkol.html )
That would be a reason to not do a mizuman over video-chat.
According to the Tzitz Eliezer, one can fulfill the mitzvot of reading the Torah, blowing the shofar, and hearing the megillah via telephone or radio. See his famous tshuva (8:11) http://hebrewbooks.org/pdfpager.aspx?req=14507&st=&pgnum=53
His permission is if there is no other way or in a case (such as a large hospital) where it is not practical to accomplish the mitzvah for a lot of people without using a microphone. So I would suggest that perhaps in this case of eating together online, since there is no other way to accomplish the mitzvah except via microphone, then it would be permitted.