yeah, it completely depends on the diode... some lasers are more efficient than others and will put out more or less light per mA than others... many diodes will burn out and die altogether at 350mA.
here's that chart senkat mentioned: http://www.laserpointerforums.com/forums/YaBB.pl?num=1210649510
when you're using a fixed-current driver, the diode will take whatever voltage it needs, the driver will keep the current regulated to what you set it to... the voltage will just sort itself out.
With my short open can running at 350mA, it had 3.1v across it.