Depends on the state. A Bill/Notice Of Sale is required in FL and it's the seller's responsibility to notify the DMV:
https://www.flhsmv.gov/pdf/forms/82050.pdf Otherwise the car remains registered to the seller and the po-po might come a knocking on the seller's door if the car is used in a crime.
EDIT: A FL "Notice of sale" (same form) does not require the buyer's signature, but fills the same requirement. So if the OP is in FL, then they can simply fill in the form and turn it into their local tax collector's office. Again, in FL the purchaser may not be able to receive their updated title from the State without this document.