Stacking freshly stained boards
Stain soaks in and dries to the touch in a fairly short time, though stain isn't a single thing. It's a broad class of products. Perhaps you're actually asking about sealers. There again, there are substantial differences in products, and your local climate play a role. 

In any case, that would depend on the expected storage duration. I wouldn't expect the boards to stick together in a day or two in cool, dry conditions. That said, the safe bet is to use something to keep them separated--raw wood strips, plastic pipe, etc.