PCB, pads with holes but with different nets on top and bottom sides
I don't see any reason why you can't create a footprint with overlayed pads on top and bottom going to different pins. Just specify the hole as unplated. If you avoid drilling one of the holes (make it an SMT pad) you may be able to avoid all DRC errors with little fuss (otherwise it may complain that the two drilled holes have insufficient clearance).  

Altium definitely understands that pads on top of each other separated by an unplated hole are not connected. 

Edit: It only seems to understand this when the unplated hole is part of a single pad. Otherwise it thinks they are connected, even with an unplated hole. 

I tried this with a simple pad on the top and a multilayer on the bottom, unplated hole and padstack edited to have copper only on the bottom and there is still a short circuit error in the DRC check, despite the unplated hole. The 3D view shows that the copper is not there, so Altium's DRC is not picking up on it.