Cocktail party problem ICA
It's not exactly mandatory to start recording at the same time, but it would be helpful.

You definitely need to be able to align the timeseries recorded by the microphones.  If they start recording at exactly the same time, then the time-series will already be aligned.  If they don't start at the same time, then you need to know exactly how much time passed between microphone 1's start time and microphone 2's start time, so you can align the timeseries after-the-fact.  Also, you won't be able to run ICA for the time period when microphone 2 was off.