The ownership of a dog can be transferred in the same manner as other property, namely by agreement, or can be transferred pursuant to animal control laws dealing with stray dogs.
Minimum requirements for contracts to permanent transfer dog ownership can be found at Adoption Organization Liability for Dog Bites. When a dog owner wants to retain ownership of his dog, but has to temporarily board it with another person for a significant period of time, issues relating to ownership, veterinary care and other special requirements of the dog can be dealt with by a written agreement such the Agreement for the Temporary Care of a Pet, drafted by Attorney Kenneth Phillips (and available upon request by contacting him through this website).
Some rescue groups make an adoptor sign a contract saying essentially that the organization retains ownership of the dog. Sometimes the contract says that ownership can revert to the organization upon the occurrence of a condition subsequent. There was a famous illustration of this a couple of years ago, when Ellen DeGeneres gave away a dog which was allegedly "stolen back" by the people who originally placed it with Ellen.
It is inadvisable to use such clauses. The organization would be held liable for dog bites in any state that has a dog bite statute. There could be other consequences too -- all of the duties of dog ownership would fall on the organization and not on the person who actually should be carrying out those duties, namely the custodian of the dog.
There have been a number of cases that determined whether the original owner of a dog has the right to get the dog back after the passage of a considerable period of time following the loss of the dog. Where the facts show some degree of negligence on the part of the original owner, and the dog's "adoption" by another person, the courts have ruled in favor of the humane society, animal control department or new owner of the dog. For example, in Johnson v. Atlanta Humane Society (1985) 173 Ga.App. 416, the court of appeals refused to allow a dog owner to take steps that could lead to retrieving his dog from the new owner of that dog, after the dog wandered away from the dog owner's property, was delivered to the local humane society, the society took steps to notify the dog owner by alerting people to contact the society if their dogs were missing, and after nine days the society lawfully adopted-out the dog. Two Vermont Supreme Court Cases provide additional analysis and apply the same principles to different situations. (Lamare v. North Country Animal League (1999) 743 A.2d 598, involving formal adoption through the local animal control authorities, and Morgan v. Kroupa (1997) 167 Vt. 99, 702 A.2nd 630, where there was no formal adoption but the new owner of the dog made a number of significant but unsuccessful efforts to alert the original owner that the dog had been found.)