You don't need a common power supply, but you certainly do need to have a common GND.
The GND is what provides the reference point for each board to determine whether signals are at zero or one (or any other level) - without a common reference, that won't work.
But the power supplies can be different - just beware of leakage if one is turned off while another is still on ...
Addenda
Sometimes you might appear to get away without a specific GND link - but it needs to be provided somehow; eg,