Caring for an animal is a 10 -15 year commitment. Remind prospects they may have to board animals during vacations and must plan for their needs before contemplating a move. Relocation is probably the number one reason that dogs are abandoned. Get 2 personal references. Even someone who does not now nor has ever owned a pet should know 2 people who can vouch for what kind of person they are.
Most vets are happy to cooperate.
If the answers are positive you can reasonably sure that the same will be done for your pet. If the preliminary screening and reference calls are positive, call the family and make an appointment for them to see your pet. Let them know you will not make any final decision until all applicants have been screened. At the interview, watch how they interact with the pet. If you and your pet like the family, set up a time to visit their home with the pet.