I cant answer regarding the co ownership.
But AKC is just a much stricter, respected, registry that require DNA testing of studs after so many litters so parentage can be confirmed if needed. Easier to keep people honest.
CKC is alot more lenient, and allow for mixed purebred registries such as yorkie poos, morkies, etc. They do not require DNA testing. They will allow registry by photos and references.
There are some very nice CKC dogs out there, and if you want a pet, trust the breeder, then registry isnt that important.
But if you want to show, breed etc......AKC is a better registry IMO.