DB Pritchard's Encyclopedia of Chess Variants calls this Tandem Chess or Double Bughouse Chess, but I've only ever heard it called Exchange Chess in the UK. The game is played with competing teams of two players, and captured pieces are recycled with 'drops', so the play remains in the middlegame.
A sample game:
Pritchard also writes about the one-board version, which requires special reversible counters to be made (and I have made a set which is in the box). It is known as Mad Mate or ChessGi, and was sold commercially as Neo-Chess. The name 'ChessGi' refers to Shogi, the Japanese form of Chess, in which drops are part of the standard game.
Playing with a partner to whom you donate captured pieces means the game can be played with normal chess equipment. A clock is essential, otherwise a player facing a forced mate will simply stop moving. 7 minutes each evolved as the standard allocation when I was at college, and I have no reason to think it wrong; 12 is good for a half-hour timetable.
Captured pieces can be dropped anywhere, except that Pawns cannot be dropped on the first or last ranks.
UK rules often include:
* no drops with mate
* no drops with check
In Exeter, we have pioneered simultaneous solo Exchange chess, played on three tables arranged in a triangle.