Yes, you can have a house and let someone live there rent free. In that case, the uncollected rent is a gift to the person who's living there, and if the value exceeds $16K per owner per year then you have to file a gift tax return. That's unlikely to be a problem in this case, since a $40K house doesn't usually rent for that much. The owner of the house does not file a Schedule E and does not get to deduct depreciation since the property is not rented for fair market value.
SSI may consider the gifted rent as income for the recipient in their calculations, so OP should investigate that before going down this path.
