I own a dog training company and I can tell you that this issues is becoming a growing problem. There are three types of dogs in this area that often get confused; Emotional Support Animal (ESA), Therapy Dog, and Service Dog.
An Emotional Support Animal (ESA) is any dog that helps someone with emotional disabilities. A mental health specialist (therapist, counselor, psychologist, etc) has to recommend that the patient needs the dog, or that they would benefit from the dog. They have to provide you with a letter from the metal health professional stating that they recommend their patient needs the animal. This animal can be something other than a dog. The letter CANNOT be from a regular doctor, it has to be a mental health professional. ESA's are not covered by law to go in any public building, they are only covered to go in the person's home.
Therapy Dogs are dogs specifically trained to help people in stressful situations such as retirement homes, cancer centers, children hospitals, etc. The dog is NOT something that the owner needs for themselves, it is something that they use to help other people. These dogs ARE NOT covered by any law to enter public buildings unless they are specifically invited to do therapy dog work.
Service Dogs - Service dogs are specifically trained to aid a person with a disability. A medical professional does not need to provide documentation showing the requirement for the dog, however, you can ask for documentation related to the disability that the dog assists with. The dog DOES NOT need to be certified in any way through any organization. This is the ONLY type of dog that can go anywhere in public, with very few exceptions.
Here are some things you can do if you suspect fowl play.
ESA - Ask for a letter from a mental health professional. If they provide it call to ensure it wasn't forged. They won't be able to give you specific information over the phone but you can verify the letter you have. Be careful of the laws if you decide to keep the letter as it is a medical document.
Therapy Dog - You're not required to admit a therapy dog at any time.
Service Dog - You can ask for the medical documentation and you can ask if the dog is specifically trained to be a service dog and if they have a disability. That's about it. If you feel they are lying you can call the ADA and talk to them about the situation. Some states have something called "Service Dog In-Training". Most of the time this allows people who are training a dog to become a service dog the same rights as a service dog. This is state to state though.
Whether the dog is an ESA or a service dog it must be under constant control. If the dog is barking or being a bother to others you can require them to address the behavior. If the behavior is not addressed then you can ask them to remove the dog. So even if they have a legitimate service dog but it is constantly barking you can tell them to correct the barking. If the barking is not corrected you can require them to remove the dog. If when they have the dog it is not under control, such as pulling them around on a leash or running up and jumping on people or even sniffing people, you can advise them to correct the behavior and again, if it's not corrected to can ask them to remove the dog. Simply put the dog should not pose a problem to anyone else around them and should be under constant control.