I co-own two horses. Both are on my property and I have the papers on both. Actually on one of them I have the "r" papers and the co-owners have the "a" papers.
I also trust the people I co-own with more than I trust my family members.
On one horse the agreement is, I cover all feed, vet, training, etc for my share. Co-owners retain breeding rights. For this horse I'm a retirement home, not that he's that old.
The other they help with expenses and I work the horses and carry to shows for their daughter to show.
Major vet bills are shared.
If I were to co-own and the horse left my physical control, I would defintely have a contract with witnessed signatures.
So much depends on how much you don't want to lose the horse. For example, if you're co-owning as part of a sale. In that case you're willing to part with the horse, if the agreement goes south, you've just lost money.
The two I co-own I have a very Charlton Heston attitude about. They'll have to pry them out of my cold, dead hands.