
I have only ever co-owned one, a male I had to home after intending to keep him. By almost 2 he started challenging his uncle for top dog and having tried to get him to settle down, he didn't so for the sake of peace in my small pack, he had to go. As I didn't know the girl who had him, I decided not to sell him outright, but to co-own him with her, at least until I saw what was what. The dog was already registered in my name, so the transfer was done in the usual way, putting both our names as the new owners. For sure all the things to do with his care went in writing - from what happened with any stud fees he might receive, to who paid the vet bills, showing entry fees and so on. In effect anything I felt could be a problem later on went down in writing!!! As it happened, his home was an excellent one and eventually, after about a year, I signed the paperwork for him to be transferred into his by then owner's sole name.
As a general rule however, I don't do co-ownership. I did have a stud agreement (one free stud) on a puppy I sold because I felt of the two males I had in the litter, he was probably going to be better for one of my bitches later on, than the male I'd kept myself. Pointless as it happened because I'd sold him to a fellow breeder (stupidly without ANY endorsements because I trusted the woman) who eventually sold him overseas, before I was ready to use him. So I used his brother on my bitch (and only had one singleton bitch puppy .... who didn't come up to scratch either).
These 'agreements' don't always work out, so for me, I make the decision about what puppies I wanted to keep, if any, from my litters and let the others go to hopefully good permanent pet homes. And I didn't ever again sell to a fellow breeder!!! If any of the puppies was good enough, later on, to show, I'd encourage the new owners to get into showing and ditto with if I wanted to use a dog I'd sold at stud - not that that ever happened either!!!
You are right - your pup would be registered in your name, using your affix. And he'd then need to be transferred into joint names. Unless the other party co-owns your affix.