I'd have to say that the fit of the rider with the horse overall, how experienced the rider is, and also how the rider as a person is. I know at my barn, besides the boarder's horses and lesson horses, the owner has a lot of horses she leases out. However, she doesn't really care too much about any of what was stated above. Most of the riders really aren't responsible, and since it isn't "their" horse, they don't really care as much. There was one girl who leased a horse across from my own and she was deathly afraid of him. Because of that she was barely ever down at the barn, and if she was, she would have to have one of the trainers there lead him for her because of her timidness.
I'm not against the idea of it, but you just have to be really careful about how the horse fits the rider and the people that you choose. :]