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In the U.S., yes, but in most other countries, no. Regardless of who trains the dog he must be correctly trained. Few pet owners would have the level of skill required to produce a true working dog.
A service dog must receive adequate training in three areas: obedience, tasks, and public access. He should be reliable in obeying commands at least 90% of the time on the first command.
He should sit, down, come, stay, and heel properly. Dogs must show manners including:
no inappropriate barking
no inappropriate jumping on strangers
no inappropriate sniffing of people.
Many people don't understand what proper heeling is, and it's actually the hardest thing to train a dog to do properly. It doesn't mean pulling a dog around on a leash. It means the dog knows where he is supposed to be relative to the person and maintains that position with a loose leash, or with no leash at all. Typically this position is next to the handler's left leg, with the dog's ear at about level with the leg and without varying more than 24 inches in any direction from that position regardless of how the handler moves (starting, stopping, turning, stepping back, etc.).
By legal definition, he must be trained to perform tasks which mitigate his handler's disability. In order to mitigate the disability, these tasks must be something the handler cannot do because of his disability. For example, opening doors for a person unable to use her hands. Any dog, no matter how well trained, who does not perform tasks that mitigate his disabled owner's disability, is not a service dog. Note that the owner must be legally disabled. That means they have an impairment that substantially limits their ability to function in things considered to be of central importance to people's lives, like seeing, hearing, thinking, walking, and using their hands.
Finally, the dog's training must be proofed and generalized so that he continues to work reliably, obeying commands on the first command at least 90% of the time when working in distracting environments, such as stores or restaurants. Proofing (distraction training) is actually the bulk of service dog training and it starts at home and at the dog training facility with distractions like food, toys and noises. Once the proofing is solid the dog is taken to various venues for generalization, or learning to apply what he has learned in different locations.
The ADA does not give public access rights to trainers. That means that unless your state gives public access rights to trainers and also recognizes private trainers or owner-trainers (as opposed to trainers for recognized programs), then you must ask permission of a store or restaurant before entering with a service-dog-in-training. See link below for information on state laws concerning service animals.