I voted 7-8, however, the last time we went on a trip to Diseny, my oldest was only 6.5 and we did not let him go in alone. He's now 7 and I'm letting him go in the bathroom by himself (or wait outside the door without me) at some places, but not all. Our next trip will be when he's 8, and I'm pretty sure I'll be fine with him using the bathroom without me by then. However, I don't think I'll be comfortable with me using the bathroom while he waits outside the door - since there are crowds that walk by the bathroom and he's gets lost easily and is the type to wander off, not realizing he's wandering off. He might mature faster than I expect, but for now, I don't think he'll be ready for that by next year. (In which case, I'd vote 9-10 for him waiting alone outside the bathroom for me but 7-8 for him going in alone).
As for other people's kids, it doesn't bother me in most cases if the child is sent in WITH a parent. A younger boy often needs supervision. An older boy by himself who is bigger than me does make me uncomforable.
*Edited to add: I just remembered, I did let him use the bathroom by himself once on our last trip alone at 6.5, but I sent DH in partway through. DH had gone to do a ride by himself close by and ODS needed to poop - which takes a long time. There was a line for the women's restroom but none for the men's. I had an 11 month old and 3 year old with me as well and the 3 year old was still finishing her ice cream. So, though I was hesistant, the inconvience was worth it to send him into the men's with DH checking in on him once he was done with his ride. He did just fine.