What is the rule for bringing a seeing eye dog on a plane?
What are the rules with seeing dogs and small areas with people with possible dog allergies?
Google to the rescue: http://www.k9man.com/flying_service_dogs.htm
I have a friend with a service dog, and she has never had a problem bringing him on an airplane. I assume that if someone seated near her had a violent allergy to dogs, they would be re-seated (she must sit in a bulkhead row because of the dog).
And this is where the Americans with Disabilities Act (ADA) is an ass. Since “service animals” are not required to be licensed or certified as such, it is left up to the owner of the animal to claim it’s a service animal. Thus, practically any animal the owner may deem it a “service animal” and bring it with them wherever they choose.
This demeans those “service animals” that in any honest sense of the word, really are trained service animals, be they seeing eye dogs, hearing dogs, etc.
Huh? Are there numerous cases of folks claiming that their pets are service animals? 
Well, there was the famous case of the pig on a US Airways flight. The woman who owned it claimed it was a theraputic companion pig that helped her heart condition by relieving stress. She also claimed it weighed 13(!) pounds, whereas it actually tipped the scales at several hundred. Other passengers were apparently Not Amused.
I’m not sure that the total could be described as “numerous,” but there have been cases in which a person asserted that an animal qualified as a “service animal” because it helped that person deal with the stress and anxiety of going out in public.
For example:
This woman and her monkey.
This woman and her ferret.
This woman and her dog.
There’s a legitimate question as to whether these truly qualify as service animals, since they’re not trained to perform or assist with any specific activity. Seeing eye dogs are trained to negotiate terrain, hearing dogs to respond in a particular way to specific sounds, and monkeys to retrieve items for a person who can’t reach them. A “psychiatric service animal” or other stress-reducing animal, on the other hand, is just there, and it’s reasonable to ask if this differs significantly from the function of any other kind of pet.
BTW, the ferret is allowed back on the bus now.
I know very little about the subject, but I know one 12 year old kid with a service dog that doesn’t “do” anything other than accompany him, but that is her job. He is high-functioning autistic, and received the dog from the Mira foundation, and since then has been able to better cope with going to school, sleeping in his room alone at night, deal with the stress he feels in regards to his younger brother (who is only 3) and generally been able to make life easier on his parents because the boy can now be left alone a bit longer while they tend to his brother or take care of housework or whatever.
I have no idea if the pig, ferret, monkey and dog mentioned in this thread are legitimate, and I agree that perhaps some sort of certification system needs to be put into place because some people do abuse the system, but to suggest that a “psychiatric service animal” isn’t as useful and as life-altering (for the better) than a guide dog for the blind, or a hearing dog, or whatever, is unfair to those people that need such an animal to be able to function in this world.
A service dog is legally defined as a dog that helps meet the special needs of a disabled person. A companion dog is, legally speaking, a meaningless phrase: if you say your Pekingese is a companion dog, not a service dog, you have no legal protection.
Additionally, in order for a service dog to be legally protected as such, the person in the pair needs to have a disability as defined by the government, which is something that significantly restricts your ability to participate in “life activities,” which of course is not defined anywhere (AFAIK). So, someone with severe epilepsy who may have uncontrolled seizures and needs a sort of early-warning/security animal to live independently has their case easily made for them. The pig is a much harder case to make.
As for the question in the OP, not a clue, but I imagine they’d have to try to find a way to accommodate both parties, possibly by putting one party on another flight.
I’ve worked with a local organization that raises & trains service dogs. All of them go to the owner with a license that certifies that they are a trained service animal, including a small wallet-sized card version of this certificate.
In addition, the owners all have documentation to show that they suffer from an ADA-recognized disability.
I don’t think that I could just go onto an airline and claim that my cat was a ‘service animal’ that kept me calm.* The incidents that people have cited here were probably in the news because the company refused to recognize them as legitimate service animals.
*In reality, it’s the other way around. I have been carefully trained by my cats to service their every desire!