Seems like an interesting idea, although the original idea of TeenyTank was to make a game that was multiplayer with just one Arduboy. As it's turn-based, the pass-and-play style makes more sense there, I think.
It's probably not too hard to do, but I am also not very time-free at the moment. The code is available at Github, if you want to have a go at modifying it yourself: https://github.com/Jezzamonn/TeenyTank/
I think what you'd want to do is to use the concept that @davidperrenoud posted above, and edit the
updateAim() function so that it either takes input from the buttons or from the Serial port depending on which player is playing. Then you just need something to figure out which player is going to be first.