I think leases vary from place to place and person to person. You might find someone who is willing to do a month to month, but most people (I assume) would rather know that their horse is going to be "in use" for 6 months+ rather then wondering...oh jeeze...is this person only going to take him/her for two months, and then I will have to find someone else to take horse.
There are a few free leases that are at my barn right now. The girls who have them found them, and do not have to pay anything for the horse, but they are required to pay board, shoeing, and any show fees. I am not sure how they have vetting arranged though.