A bank can also act a de factor escrow service. Banks do this all the time and will walk you through the process.
Scenario A (No Liens)
1. You are financing the car purchase with your local bank or credit union.
2. The seller's car does not have a lien on it, no loans, title in hand.
3. You apply for the loan and get qualified.
4. Upon qualification, the bank will request documents from the seller (copy of title, registration, driver's license, etc).
5. The bank will run a title search on the car to verify that there are no liens on it (it's their ass on the line if there's a problem).
6. Bank will request both parties sign paperwork. This can be done together or at each party's convenience.
7. Seller signs the title to the bank.
8. Bank will issue a cashier's check to you and you hand it off to the seller.
9. Seller gives you the keys.
Scenario B (Loan or Lien on Car)
1. You are financing the car purchase with your local bank or credit union.
2. The seller's car a lien on it. Title is held at the seller's bank.
3. You apply for the loan with your bank and get qualified.
4. Upon qualification, the bank will request documents from the seller (registration, driver's license, etc).
5. Bank will also request documents from the seller's bank (Payoff amount, copy of title, etc).
6. The bank will run a title search on the car to verify that there are no additional liens on title.
7. Both banks will request signed paperwork. This can be done together or at each party's convenience.
8. Your bank will send funds to the seller's bank. They can help arrange additional funds. For example if you're buying a car for $50K, the seller owed $30K, you would need to provide an additional $20K to complete the transaction.
9. Upon receiving funds to pay off loan, seller's bank will arrange for the title to be taken care of. You would probably sign the title, but it would then go straight to your bank.
10. Seller gives you the keys.