I imagine it's just like running two amps on an ABY, but with dirt pedals, right? There maybe ground loops or signal loading, but you may as well try it the easy way first to see if that's an issue. It's probably really cool. I've never done it. I'm tempted to try it with a stereo pedal and see what happens. But I think it would only work with the effect engaged, and I wouldn't want that, at least at first. Try it. It won't hurt anything.
Unless there is a buffer in the splitter you are going to get major tone suck. Think about the Ernie Ball VP Jr and plugging in the tuner. I split my signal a after the dirt pedals to run delays in parallel through a Radial Bight. Levels had to be adjusted when both signal paths were active.
I've thought about building an overdrive with two parallel gain stages, and someone informed me that if the gain stages are out of phase, cancellation would occur when blending the signals back together. (Inverting and non-inverting feedback loops on opamps)
My initial thought when you posted this was "cancellation" but then I realized you wouldn't be blending the two signals back together. They're going to two separate destinations, so...I agree, I think you could get some really interesting sounds.
It's very cool! I think all of the considerations have been mentioned here. I'm not a fan of mixing clean with dirty, it sounds too layered and weird to me, but fuzz + OD, or 2 ODs sound great. I think the best use of 2 amps is with stereo modulation pedals. Stereo reverb, flanger, etc. sound very 3D. You can set up 2 amps with a delay so one amp gets the un-effected signal and the other amp gets just the delays.