Is there any person who is widely believed to be enlightened by Buddhists, and who confirms that himself/herself?
I cannot comment on the other answers yet due to lack of "reputation points" on this site, but to add to Crab Bucket's answer, yes, Daniel Ingram is a a well known taboo-buster in the subject of claiming enlightenment.
As of today (1/17/2015), he still claims arahatship, he never rescinded his claim.
During the genesis of his (Ingram's) original community (the Dharma Overground), there were a lot of practitioners who made strides in attaining varying depths of awakening. A lot of them went on to spawn their own communities and do their own thing.
He originally started a community called the Dharma Underground, which was meant to discuss hardcore meditation practice (hardcore meaning no nonsense and down to earth), but later decided to open it to the public as the Dharma Overground!
The original Wetpaint community had about 500 people but it multiplied exponentially producing all sorts of awakened individuals.
Check out Buddhist Geeks for sure, I'm also pretty sure hardcore (regular) meditators at various retreat centers also make good progress in becoming awakened, but often do not recognize their progress due to unrealistic models of attainment, and plus a lack of willingness to talk about maps and models.
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.
I am reminded of the saying: "Those who know, say not. Those who say, know not." If I say "I am this" or people say "He is this" then I am further bound into the snare of identifying with the human experience. If you are aware of your breath and you are in awe of a sunrise, what is to say that you have not awakened in this moment...even if in the next moment there are worries and concerns crowding in? That is why the Mahayana school emphasizes seeking the enlightenment or reducing suffering of all so that the yoke of self seeking diminishes. As Roshi Shunryu Suzuki says about the practice of meditation, we are both mahayana and hinayana. This strikes a chord within me. We seek enlightenment but to whatever degree we experience we dedicate it to the benefit of all sentient beings. Thus we are not trying to attain a goal to go somewhere we are not, but glide on this river of peace effortlessly moving on the river of being back to where we have always been. This feels like a big burden off the shoulders. It does not condone laziness, but the meditation extends beyond sitting and the sitting is something one would not give up for something in this world.
Out of the 7 billion people on this planet. there should be thousands if not more of enlighten people, some of them were born that way, others became awaken spontaneously (like the Buddha), and others became awaken after a long spiritual training (such as in zen and Buddhist and kabbala) training. Most of the enlightened people do not call themselves enlightened or keep it a secret except for a few students. One of the persons which do publishes his enlightenment is John De-Ruiter. After meeting him in person and reading his stuff, I think he may be enlightened. Unfortunately, the only way his suggested for other people to become enlightened is is to be around enlightened people as much as they can.
I'm not saying that there is or isn't, but concerning the self-confirmation of such, this is not so straightforward as it may seem, considering a system where enlightenment is concerned, in part at least, with the transcendence of conceptual self-hood:
From Chapter 9 of the Diamond Sutra: “
What do you think, Subhuti? Does an Arhat think like this, ‘I have attained the fruit of Arhatship.’?” Subhuti replied, “No, World-Honored One. Why? There is no separately existing thing that can be called Arhat. If an Arhat gives rise to the thought that he has attained the fruit of Arhatship, then he is still caught up in the idea of a self, a person, a living being, and a life span.
While it is true that the sutras tell us that the Buddha attained enlightenment, it is worthwhile to remember that when he first declared this that he was rejected by Ājīvaka Upaka, who just shook his head and walked away at this bold claim. When he later got to Deer Park and the monks addressed him by name, he did not right out declare arahantship, he said instead that that is not the proper way to address the Tathagata. This subtle, and non-trivial difference reflects the fact that he never went out "to declare", he went out to teach, which is clear, as is revealed the well known story of the conversation with Brahma just post attainment. The difference at play is about the primary intention behind the activity, teaching for the sake of others (to an appropriate audience) vs. making a self-declaration.
Remembering that these monks in Deer Park could not resist attending to him, after he instructed that it is improper to address the Tathagata by name and as friend he simply made statements like, "The tathagata is this... the tathagata is like that," and avoided speech like, "I am this, I am that," He does say, "I will instruct you," But this is less of a self declaration than a promise of activity.
So, to be direct with your question, coming out as the tathagata was, in fact, a problematic situation for the Buddha. It was one that was met and conquered, but it was still problematic.