This varies a lot based on the kind of community this is about.
For non-professional communities - say, random web fora or similar - the name should always be taken as a pseudonym. As long as the user isn't actively impersonating someone (and, even then, the usual line is drawn at impersonating moderators), usually no action is necessary.
Of course, if someone is actively impersonating, contacting them and asking them about it might be a good idea, but I'd still largely leave it be with said notion of pseudonyms.
Professional communities, on the other hand - knowledge bases, industry fora, things like that - can have a problem with impersonation. Here, confirming the identity may be crucial. The tales of Jon Skeet and Jeff Atwood users on SO are astounding, and that's not even that professional a site.
As for how to confirm: Obviously you need some form of end point that you can definitively note as belonging to the person you're trying to identify as. Be that an email account (although, SMTP shenanigans and all ... you know what, don't use email for that.), a twitter account, or the IT favourite GPG signatures - these all work, as long as you can confirm that these are actually validated by the "notable person".
If you cannot identify the user as said "notable person", but your community requires it, you need to do two things - and you need to apply these throughout the forum, regardless of the "fame" of a given person:
- Have a rule set up that confirms this requirement.
- Anonymize the user until identification can be provided.
If the user cannot identify in a sufficient manner for the requirements of the given community, their account must use a different pseudonym. Let the user pick a new name, at least temporarily.
(This all of course also doesn't address the matter of "Hey, I have the same name as that guy!", but it's a good rule of thumb that, if laid out well enough, should be sufficient for most instances.)