It is unknown if that's possible for most of the people, but Cornelius Fudge, then-Minister for Magic, was known to have visited Sirius Black once (Prisoner of Azkaban).
It is also known that Barty Crouch and his wife were granted a deathbed visit for their son.
Probably only people in High ministry post are allowed to visit prisoners not ordinary wizards.
Probably not so. Arthur Weasley had visited Azkaban once as a Ministry official, and he's not that high ranked, is he?
Maybe it's only people of the Ministry of Magic. Ah.. I love the mysteries of Harry Potter.