You MUST join the grounds together. You can't have one board floating with respect to the other
with some kinds of isolation you can actually have two boards communicating with no galvanic connection whatsoever. That this method, for this purpose, would be a convoluted approach, is another story. For instance there are modern 485 transcievers that can hand quite a bit of common mode offset (I know of 60V, there may be better chips available)

with the same ground plane
NO, that would not be good; however with the same layer (typically #3) split in two and connected at the right place used as ground plane for both is doable.

Erik