This is basically the same question as Is progress in the coop campaign in Portal 2 saved?, but that one is PC-specific and I'm asking about the Xbox 360 version. The major difference is that there's no such thing as a Steam Cloud on 360. So, is this possible, and if so, how is it done?
Once you complete a test chamber the game marks it as completed by you forever. When you return to the "hub", you can access all the levels of the player with the most progress.
For example if P1 has just started and P2 is up to chapter 3, you can play all levels up to P2's chapter 3 progression.
You can continue a chapter from test chamber by selecting the level on the control panel that opens up in the wall at the entrance to the re-assembly units (the elevators) for each chapter. Press the big arrow keys (with the use button on your controller) to move the black underline to select different levels, or the next level you haven't completed in the chapter. Enter the re-assembly units to start that level.
Any progress you make within a test chamber is not saved.
Okay, I just took a chance and quit. Turns out the game silently saves your progress in terms of testchambers completed. If you're in a testchamber when you quit, though, you lose the progress from that chamber.
For example, I started a new game and played through Course 3, Testchamber 3. When I returned to the Hub after quitting, the game remembered that I had passed everything up to Course 3, Testchamber 2, but I had to start chamber 3 over again.