I stay if it is at a public place (mall for Build A Bear, Chuck E Cheese, for example), but I drop off if it is at someone's house. I'm just not comfortable that one or two adults can watch 10 or more kids in a public place like that. I'm on the fence at a closed in public place, like Pump It Up, which is a room of bounce equipment and then a different room for the cake. I would hang for a few minutes to make sure that she understood the setup and then go. DD is in the first grade, and I've been doing that since last year.
I just read the post about the pool - I would never have dropped DD at a pool party at 4 years old! I'm not sure that I would do it at 7 years old, and she can swim fairly well!