Are there any living enlightened people? Is there any person who is widely believed to be enlightened by Buddhists, and who confirms that himself/herself?
As Andrei says in the comments talking about achievements are taboo so it's difficult to know. However Daniel Ingram (Theravada practitioner) is a taboo buster about this and openly claims in his book Mastering the Core teachings that he is an ahrat. So basically that he is enlightened. 

That might seem like massive hubris and just in bad taste but if you listen to any talks by him he is massively engaging on the subject and convincing. I recommend listening to an interview with him on the Buddhist Geeks podcast (numbers 6, 9 and 11). In that talk he claims that there are a significant number of people either enlightened or well on the way. He names some American monasteries and particularly the Insight Mediation Centre as a place that you might bump into these people. He claims that although no-one is going to admit it publicly, if you a a serious practitioner and physically go there with particular issues or experiences of your own then those people will be more forthcoming on their own experiences. This accords with my own limited experience. In a well established relationship with a teacher or more experience practitioner then people will be a lot more forthcoming.

As a further interesting point, Daniel Ingram does say that our models of enlightenment are a mess (he's a provocative kind of person). So when we are casting around and trying to spot who is further down that path then we might just be looking for the wrong kind of things. He particularly says that being enlightened and being a very saintly kind of person wouldn't necessarily be the same thing.

As a note - I don't think Daniel Ingram still claims he is an arhat (not sure). He has a new edition of his book coming out so it will be interesting to see where he is at. Also I'm not necessarily saying Daniel Ingram is right, but just that he has some very interesting views in this area and I frankly like writing about them.