How can you buy a car from a person who still owes their bank?

Yikes. I hope you're getting receipts from the seller. The cleanest way to do this deal is, to apply for a loan from a bank and do the transaction through the banks. Or, contact the bank that holds the note to see if you can assume it. If the seller defaulted on his payments, the bank would come get the car that you've been making good faith payments on. As this deal stands, you don't have much protection, or recourse, if the seller isn't making the payments to the bank, or defaults, later.