There is no single set of requirements for dog fitness training. It depends on many factors, including the country where it is offered and the requirements of the employer.
In some countries, there may be specific certifications that one could acquire to work as a dog fitness trainer. For example, in the United States, there are several organizations that offer certifications for dog fitness trainers, including the Canine Fitness Coach Certification Program and the Certified Canine Fitness Trainer Program.
Some dog fitness trainers have a veterinary background, some are trained in dog training, and still others have extensive experience working with dogs. However, there are no legal requirements for canine fitness trainers, so we always recommend thoroughly researching a potential trainer's background and experience before deciding to work with them.