I've not noticed anyone else mention using multiple microphones, but with two microphones spaced a few metres apart a computer can separate out two people talking over each other due to speed-of-sound differences (i.e. conversation A reaches mic 1 then mic 2, conversation B reaches mic 2 then mic 1).
I've seen two people talking separated using two microphones, I don't think you need more microphones for more people. This is a common machine learning demo (how I've seen it), but it is just signal processing really (i.e. no machine learning required).
I'm not aware of any products integrating this however, but I've never had the need to look. Is there such a thing as a bluetooth microphone for iPhone/Android? Like a headset but without the speaker. If you're a programmer maybe you could write an app that separates out different audio "streams" and then send each to existing voice recognition APIs.