Service dogs in training are not guaranteed public access by the ADA. Only task trained dogs (and the rare miniature horse) WITH a disabled handler or the disabled person plus a trainer are guaranteed public access.
ESAs are not guaranteed PUBLIC access by the ADA, but must be allowed in housing (unless shared with host) by Airbnb's rules, FHA and some airlines (although airlines are tightening that up.) ESAs don't have to be TASK trained but they do have to be trained. The ESA handler actually must have a disability.
The handler does not have to be permitted to leave the dog at the listing unattended (ever.)
We do allow dogs anyway but would not appreciate being forced to allow fake service dogs/fake ESAs.
The person above who says they can bring their puppy in training with guaranteed access is incorrect by the way. Access is not guaranteed for dogs in training under ADA.
Psychiatric SERVICE dogs have guaranteed access. They are TASK trained and must initiate the task without a command if the handler's disability at times prevents them from giving a command. The parent of the child or the assistant of the disabled person can by proxy give the command.
ESAs only have guaranteed access to the limited situations of housing and air travel but again only if the handler has a DISABILITY. If the handler is not disabled then they don't have access with their ESA.