Legally, all that is needed for a private sale is for the buyer to obtain a permit before the transaction occurs. The seller is under no legal obligation to make a copy of the permit, retain the permit, or otherwise have anything to do with the permit.
Personally, if I'm the seller, I'm going to make a bill of sale...but I would do that for a lawn mower, a motorcycle, or even a cool knife or something. There's no requirement for it, though. It's just something I'm going to do. If you'd like to make a copy of the permit and write out a bill of sale for your own peace of mind, it would probably be a good idea. But, that's up to you.