You don't need special training to offer dog fitness. However, if dog fitness trainers have a background in veterinary medicine or training in dog training, of course, it can not hurt. However, it is also enough to simply have extensive experience in working with dogs.
In some countries, there are certain certifications that can be obtained in order to work as a dog fitness trainer. For example, in the USA, the organizations that offer certifications for dog fitness trainers are called, for example, "Canine Fitness Coach Certification Program" or the "Certified Canine Fitness Trainer Program".
However, it is always important to know that there are no legal requirements for dog fitness trainers and therefore it is always advisable to check the background and experience of one carefully before deciding to work with them.