You could play multiplayer Tetris that way. I think I saw it once in my life.
There‘s also an excellent Gameboy Color romhack of Dr Mario that supports multiplayer. Recently tried that out with my girlfriend and it was a lot of fun.
Edit: this is it for anyone interested. Looks like even the original version for the Gameboy supports multiplayer.
I played multiplayer Tetris frequently.
When you get lines, your opponent’s stack pushes a line with a gap up from below, except when you get a Tetris, which pushes four lines (with the gap aligned, so you could Tetris back and forth).
You had an indicator for the max height of your opponent’s stack next to yours.
Great game.