Why is it acceptable to cite unpublished works? If an article, theorem, etc. is unpublished, presumably the work is only available to people who track down the author and get a draft copy. Today we have arXiv and can find many preprints on the Internet, but people have been citing unpublished works longer than arXiv has been around. It seems like an unscrupulous author could claim anything he wants, cite an unpublished work that doesn't support his claim (or even one that doesn't exist), and get away with it unless the bad claim is caught in the review process.

I suppose the reviewers could ask for a copy of the cited work, but I recently saw a citation that was just "private communication 1981" for a paper published in 1985, and that could easily have been a phone call where no record could be provided to a reviewer.

I've never published anything myself, so maybe I'm unaware of a part of the review process that is meant to catch this, but what prevents authors from making bad citations of unpublished works?
You seem to be interpreting citations as a way of proving something by appeal to authority.  That is not what citations are.  A citation is a way of giving appropriate credit to the originator of an idea and pointing the reader to the explanation of that idea.  Not giving credit to the person who gave you an important original idea is unethical, whether that idea has been published or not.

Citing a "personal communication" related to a mathematical fact (for instance) does not eliminate the need to prove that fact.  Appropriate usage would be to write something like "the following was communicated to me by X. Y. [personal communication]", followed by (for instance) a theorem statement and its proof.

Note that if something is in a technical report or on arXiv, it is not unpublished.  People publish blog posts, novels, and Youtube videos.  Published and peer-reviewed are completely different things and it's wrong to conflate them.