Once you have sold your horse, you no longer have control over what happens to the horse. A good home may no longer be able to keep a horse due to death, disability, divorce, unemployment, job transfer, etc. A horse is considered property. 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.