Voluntary registration is different from a firearm transfer errantly often referred to as "registering". When you perform a normal transfer, the DROS is sent to the DOJ, for the purpose of a background check during the waiting period. They by law are not allowed to keep ANY RECORD of this information. The only "paper trail" when transferring a firearm in CA is that of the DROS kept on file at the point of sale.
A voluntary registration will be kept with the DOJ, and is often a form of proof of ownership used for insurance purposes. If you friend "gave" you the firearm without a transfer through a licensed dealer, that is illegal, unless it's a family member, or an antique. I suggest a legal transfer. Perhaps the term should be an extended "loan" until you finish the paperwork. Since he wants a transfer, that is indicating he currently holds interest in said firearm. I'm not a lawyer, but I'd call that a loan.
The information I am presenting is yet another reason to join CRPA (California Rifle and Pistol Association) if you are in CA (or if not, too!
) as they send you out a laymans law booklet.