Justin, you've got the right idea. I would use a dedicated power supply just for the peace of mind that the monster heatbed isn't going to trip a fuse on the control electronics or something of that sort. It's possible.
Mechanical relays can't keep up with PWM (pulsing on and off at some moderate to high speed, which is how the target temperature is maintained in all the heating elements). What you'll want, ideally, is an optical isolation relay, also referred to as a solid state relay. A transistor or mosfet would work as well, but I'm recommending to completely isolate the 2 circuits from each other.
Quick primer: http://arduino-info.wikispaces.com/RelayIsolation
From taking a look, it seems that an opto isolator relay of the amperage you are requesting will be expensive. Where did you get the 50 amps number from?
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