Yes, you must fill in the relevant sections on the FAC of the buyer (if it's a private sale). Otherwise if it's going to a RFD they will enter it onto their books. You should then inform your FLD (if memory serves within 14 days) of the transfer of ownership giving details of who its gone to etc.
Most forces have a standard form that you can fill out and send in, to make sure you send in the correct information.