My 15-year old had a negative experience with this just yesterday:
She was invited to a close friend's 16th birthday party, and she was very excited to go. The invitation just said "dinner at X restaurant". That's the most typical type of Sweet 16 around here -- just a casual dinner with friends. It never occured to me for one moment that the girls' parents weren't going to attend or that guests were expected to pay for their own meal. My daughter bought a very nice gift for the girl (nice pair of birthstone earrings), wrapped them nicely, showed up on time. The birthday girl was late, no adults attended, and -- surprise! -- each teen was supposed to pay for her own meal. Thank goodness my daughter had a few dollars in her pocket (we talked about how she'd better be prepared for this to happen again; she shouldn't go out without enough money for a meal). It wasn't really a "party"; rather, it was just people getting together for dinner and being tricked into bringing gifts.
Two FYIs: These kids are all 15-16. The birthday girl's family doesn't seem to be short on money.
When I picked her up, my daughter was a bit miffed about it. She defended her friend saying, "It's not a big deal", but I
don't think that's how she really felt. Although I like this friend, I told my daughter that her friend's behavior was rude. She's been to plenty of birthday parties, and she's never needed even a dollar to attend.
I mentioned it to my husband later, and he was aghast. He said -- and I agree -- that he'd be mortified to invite someone to a party and then ask them to pay.
And then I realized that this is going to be a problem next week too: It's birthday season around here, and my daughter's also turning 16 next week. We're doing a very similar

Dinner at X restaurant (different place, but also a casual restaurant). We've reserved a room in the back, my husband and I will be there to chaperone (though we've arranged a big table for the girls and a smaller table to the side for us and a few other adults), we're bringing a
very nice tiered cake, and we're
definitely paying for everything. My daughter also made some pretty nice gift bags for the girls to take home. The last-week-birthday-friend is invited . . . I wonder if she's going to be embarassed. If she is, I can't help that. I would never do anything to bring attention to her last-week party, but since there's large overlap in the guest lists, people are going to notice.