I have tried the highest graphic settings (can't remember what you called it
Something fun, though ('fantastic')) and 'good'. It happened in both.
My videocard is a Radeon HD 5770.
So far every time I've tried the game (5/6 times) it has happened, all but one of those times to me (the other time my partner stopped moving and eventually left - but stayed around for about a minute and a half, so I'm guessing he/she might have gotten the same bug).
The first time it happened after the first puzzle (with the circle and square) - immediately after that my part of the screen went black, I could see my partner continuing.
One of the times I actually did see the dual screen with both of us moving, and we got a bit further (until the part where you first have to turn your head to look to the right, but my partner had just left at that point so I quit).
Other times I've had both of our parts of the screen go black (with the emoticon icons still visible and clickable), once I've had mine go grey and the partner's part black or the whole screen went black (full screen) but I could still move (could hear sound of moving/jumping/down arrow).
Also - at the very beginning, if you're playing red, is it intended that you are not be able to see anything until your partner reaches the first 'window'?
Edit: It always happens at a moment after we solve something and the camera changes in one way or another.