The problem won't be ground loops, it'll be load sharing.
There's absolutely no need to separate the grounds. You won't even be able to to; they'll be connected to the PC chassis, the case of the PSU, and mains earth.
What you do need to be concerned about is what happens if you connect both supplies' outputs together, either intentionally or through one of the things you're powering. Chances are they're not designed for it, and won't like it.
The danger is that, suppose one of them has a "12V" output set at +12.07V, and the other is set at +11.92V. At what voltage does the combined output sit, and how much current comes from each supply? I'll give you a clue, it won't average out to +11.99V with 50% of the current coming from each... they'll fight each other, could go unstable, and might even smoke.
Get one power supply that has a high enough rating to run your whole PC.