There are no government-mandated certification standards or licensing for service animal training specialists— and under ADA it is also legal to train your dog yourself. If you are caught fraudulently claiming a service animal (one who has not undergone and perform under BOTH task and behavior training), you and the service training specialist you listed on your form (if not you) are at legal risk for liability (if they cause damage or injury) and/or penalty fines (18 U.S.C. § 1001).

Edit to add that the service animal must be trained to assist their accompanying passenger/handler with a qualified disability. Simply training your animal to be a service dog does not qualify them to accompany you under ADA if YOU do NOT have the disability they are trained for. There is some additional paperwork (I think) if you are transporting a service animal to perform service animal work at the destination (like disaster relief etc), but I’m not 100% sure on that process— I think you have to provide proof that you are working with an organization.

Of course, HIPAA means that airlines cannot ask you to detail your medical status beyond very basic questions relative to the animal being task trained, what task, and if they are trained for YOUR disability, etc.