I have had friends haul my horses on several occasions. I never even thought of getting any type of contract. My horses are covered under my home/farm owners policy.
If you are worried about it you could just type up a simple contract stating that you (the hauler) are not responsible for any injury's or possible death to the animals. Likewise they are not responsible for any damage their animals could cause to your property (trailer) and everyone signs it.
"Until one has loved an animal, part of one's soul remains unawakened..."
- Anatole France