I have moderate social anxiety and agoraphobia as well as generalized anxiety. The dog I raised from a puppy is a complete social butterfly, excellent around all people and dogs. The adult dog I got from the Humane Society, which came with aggression issues, has become a solid, dependable animal. He does have a definite "bubble" where strange dogs are concerned (particularly terriers) but it is predictable and getting steadily smaller.
I think part of the issue is how the owner handles themselves. If you let your anxiety keep you from getting out and you don't manage to socialize your dog or work with it on a regular basis, you may have problems. If you are unable to control your reactions on a regular basis, you may have problems. But if you are actively working on both yourself and your dog, I believe it can work.
Actually, I am better since getting my dogs. They offer something to focus on when the world is overwhelming. I also discovered that I really like being around dogs, and training them, and talking about them. It gives you another skill set, which can be helpful in building confidence and self-worth.