You need to log the firearm's serial number, seller, importer (if of foreign manufacture), caliber and type (rifle or handgun) into your C&R log within 24 hours of receiving the firearm from your shipper. That's it.
At some point, you may have to show your C&R firearms and log book to an ATF agent - The ATF seems to have quite a bit of extra time on its hands right now, so "audits" of C&R collectors are not unheard of. As someone who has been recently "audited", I can assure you that it's no big deal.
ETA: If the seller also has a C&R license (and the firearm is C&R eligible), he is required to keep essentially the same info his his logbook.
That’s the catch - I had forgotten about the logbook.