I would expect the bar owners to understand the risks and ensure that customers don't drink the stuff, which they appear to have failed to do. It's only going to be present very briefly, unless they're adding loads, so this wouldn't usually be a problem - AIUI the girls friends brought her the drink which she promptly drank, as you would at your party.
Liquid nitrogen's not terribly bad unless it's trapped against your skin (or innards). In which case it's pretty nasty - it could be compared to having 250C hot oil in one's drink. One can dip a hand up to the wrist in it as long as it's promptly removed, and not feel anything more than a cool sensation. I don't generally use gloves when handling LN2 as splashes just bounce off skin - the only injury I've witnessed from it was when some got trapped in the fold of a glove, resulting in a burn blister.
I'd agree that dry ice is better in drinks though - it sinks and thus makes a more impressive bubbling/mist effect - LN2 floats on the surface and thus doesn't generate as much mist. It's also easier and safer to handle - though probably just as bad if ingested.
