Until you get to the mini-boss and boss monsters, that is. Many of them literally take up both screens, and even those that don't wear a size 347 shoe tend to roam around upper and lower screens at will. However, each is pattern-based, like in an old arcade game, and once you've learned their patterns, those bigger-they-ares can be harder-they-felled with ease. It helps if you use your rage attack, a screen-clearing super-move that works even better if both monsters use it in split-second succession.
Above: Only four of the 10 playable monsters have been announced, not to mention the creatures playing the villain roles
Speaking of both at once, now's a good time to clear up exactly how this happens. When playing with two players (even if there's only one cartridge, thank you developers), both monsters are onscreen at all times, with each player controlling one or the other. When stamping through solo mode, you usually only have one monster onscreen, but you can swap between them at will (as long as you're not performing an attack move at that exact moment). So that combo-super move is possible even if you're playing alone. Which is nice, because even if Atari hasn't technically announced the huge mutant rose bush or massive lobster you're trying to shred, you'll be well prepared to kill them.