Contracts drawn up by attorney. Obtain Social Security and driver's license numbers of parties entering contract. People can and do move. Phone numbers change. Con-artists have entered into written leases and then disappeared with the horse.
People are more apt to follow the rules if there is a contract.
Law enforcement considers breach of contract a civil matter. Expect to go into court yourself or be prepared to hire an attorney if the person leasing your horse breaks the contract. It is more difficult to enforce a civil contract across state lines. Experienced horsemen know how much it costs to maintain that horse. They may have a high maintainance horse on their farm, but that is because that horse is special to them .