Any ideas where I'm going wrong?
Well, you didn't tell us what the problem is.
Open the Serial Monitor, instead of Processing. Do you see the 0 and 1 appear there? Probably not, since 0 and 1 are not ASCII characters. So, the question becomes why are you sending the values as bytes instead of characters?
What does your Processing code look like? It's difficult to diagnose a sender/receiver problem when you only post the sender code and don't describe the problem.