Mistakes smuggled out by employees are called printer's waste, not recognized as legitimate stamps, and may be confiscated from collectors; the Nixon invert is a well-known recent example of an apparent new error that turned out to be simple theft by insiders.
The authorities may attempt to lay hands on legitimately sold errors, as happened with the original Inverted Jenny sheet, but usually, collectors are smart enough to hang onto the windfall.
Genuine errors are uncommon or even rare; postal administrations have several layers of quality control and inspection, and most printing problems are addressed before the stamps ever reach the public.
The usual sort of oddity is misregistration on a multi-colored stamp, which can result in shirts apparently with two sets of buttons, eyes above the top of a person's head, and so forth.
The Canadian Christmas stamp of 1898, depicting a map of the world with British possessions in red, is famous for unusual color oddities that appear to claim all of Europe, or the United States, or central Asia for Britain.