That varies by state. Where I live, the financing institution with the lien keeps the title until it is paid off.
Kinda nice when you trade in a car you still owe a bit of money on, as you don't really need to bring anything with you other than the key and your drivers license; the dealer does all the work getting it cleared and transferred.
Source: Trying to get a bank to remove the lien after it’s fully paid off can be a real source of frustration.