If the distance from the microphone to an "unwanted source" is three times the distance asthat from the microphone to the source phasing likely wont be an issue.
There's always caveats with engineering but it's a decent rule of thumb assuming equal volume sources... I can imagine it's not too hard to detect that anyway, weve been able to do realtime fft for a very long time.
No, probably not there are no phase issues if you just don't transmit the signal. The hard part would be to determine who's in the room, and then who's talking and then mixing appropriately to eliminate feedback and optimize speaker sound quality. None of which requires signal phase accurate synchronicity.
If they're actually able to "sync" (again a poorly defined term) given the problems associated with network latency and different hardware it would border on magic.
