I would absolutely not co-own a pet dog. For any reason. I wouldn't want my signature on a piece of paper that may give the breeder any chance of taking my dog away. Even if the agreement stated conditions, etc., I would worry about it being argued in court.
The registered co ownership is something that is common in show and performance circles. Its very common in conformation especially.
My dog Merlin that is in my avatar is co owned by the breeder. What is on the AKC papers is not nearly important as what is in the purchase contract.
If it is spelled out and everyone understands the contract their part of the bargain.
My contract clearly states that the dog is to live, be trained, cared for, etc with me for life.
That I am to make my best effort to show him and attain his AKC Championship. AS TIME AND FINANCES ALLOW. There are no time lines, no demand that I have him in the ring every weekend. etc.
That once the dog acheives his AKC Championship and is retired (my choice when he stops once the championship is attained) that full ownership is signed over to me.
I am to have the dogs eyes, ears, hips, cardiac, patellas, elbows tested at my expense with a vet she approves of.
The breeder at her discretion has the right to use the dog at stud, once his testing is complete, to any tested female of appropriate breeding age that she owns or co owns. This ageement is for the life or which time he becomes unable or unsuitable for breeding.
I am free to breed the dog, once he has satifactory completed his testing, to any female from any line so long at the female has passed all genetic and health clearances. Any compensation in the form of stud fees, litter picks, etc are retained by me.
IF at any time, the dog is shown to be unsound or less than represented in reference to genetic soundness, he is to be neutered at my expense, and retained by me as a pet. At such time as this occurs the breeder will replace this dog with a specimen that she and I agree is of the quality the original dog in question was represented to be.
Now what does this do for me?
He is my dog, he lives with me, he is cared for by me, etc.
I have assurances that if it is ever shown that he is not the genetic specimen I paid for, then I get to keep this dog, and upon neutering, will receive a replacement animal.
What does it do for the breeder?
She has assurances that I will make an effort to put his AKC championship on him.
She retains access to his Genetics should she ever decide to use him.
What does this do for the dog?
Well if I or someone she was comfortable had not purchased him under such a contract, she would have kept him. And he would have had a good life. But he would have shared time with numerous other dogs, There would have been tighter controls on his free time. As she has a couple of other intact males on the property. Care and rotation would have had to have been used to avoid serious conflicts between the males.
But with me, he gets nearly all of my attention. Gets to go lots of places, meet new people, have fun, does not have to compete with other dogs for my attention, (I have another dog at the house, but he is my wifes dog. He is more devoted and bonded to her and my dog to me. But both get attention from both of us) The other dog is a neutered male, and he is easy going and has no desire to compete with my dog, so no conflicts. They are buds.
I see this working out for everyone.