I've noticed that more and more often people at informal gatherings are liable to introduce themselves by given name only, presumably because that seems more friendly; but if you want to establish any kind of ongoing connection you’ll need to provide a family name as well. There are times I have suspected that the other person is thinking “I don’t expect to ever see this guy again, so I’ll just go with my first name.”
Once the pattern is established, it’s awkward for a later speaker to give his or her full name instead—though that might be genuinely useful, especially if one anticipates working on a project with the new acquaintance.
The Japanese generally exchange cards upon meeting, which seems very formal to Americans but can be quite useful.
In a purely casual social gathering—such as encountering someone at a bar—one person may want to preserve her/his privacy by going with given name or nickname only, whereas the other person may hope to establish an ongoing connection by offering their full name. I see no way around this except to be conscious of what each pattern may imply.
If you intend your new acquaintance to get in touch with you, it’s best to go with full name. The same goes for praising individuals in a public speech, where you should try to make clear just who it is you’re talking about if not everyone in audience knows the individuals already.