How would you feel if the girl you see from time to time came up to you while you were with your family?
Just think how SHE would feel if you recognized her, and god forbid, approached her. She could be with her family, friends or on a date with a client. Thankfully, I have never been put in this kind of situation, but I can say right now that I would be absolutely mortified - especially, if they called me by my stage name. Yes, I have heard of this happening.
After you fantasy ends we go back to our real lives. There is a delicate balance we have to keep between our private and professional lives, so all our efforts would be for naught if a someone forgets himself and thinks that it is okay to approach a provider outside of the fantasy boundary. Sharing a look and a smile would be considered tame and appropriate at the right time, but anything further might be pushing it.
Some girls might be okay with being recognized and approached in public by past clients, but speaking for myself: NO. Playing it cool would be the best policy, and you can laugh about it later when you see her again.