If I saw my doctor/dentist/barber outside of their usual places of business then I would smile, say hello, possibly ask how they were doing, and then continue on my way. I wouldn't buy champagne, even though my dentist happens to be really hot. If I ran into an escort I knew outside of her usual place of business then I would smile, say hello, possibly ask how she was doing, and then continue on my way. Assuming, of course, that neither of us was with anyone who might ask awkward questions about how we knew each other.
“I wish you wouldn’t keep appearing and vanishing so suddenly; you make one quite giddy!”
“All right,” said the Cat; and this time it vanished quite slowly, beginning with the end of the tail, and ending with the grin, which remained some time after the rest of it had gone.