There is a type of chess that is called something like "doubles", where two people play against another two and when someone takes a piece from their opponent, it goes to their friend and they can put it anywhere on board as long as it is their move. (On a move you either put a piece, if you have one or make a regular move) However some extra modifications would also be necessary - you can put pawns only from 2nd to 6th rank and when you are in checkmate and you have a piece you can put to block it you can do it. Also as soon as one player is checkmated their team loses. Ideally you would be able to see both boards and pieces you can put on your board. Also timing would be useful.
I am sure this would quickly become popular but it would need coding "a bit".
I am sure this would quickly become popular but it would need coding "a bit".