But if the belief is accurate either way, then you can basically pick whatever belief you want. This is the weird paradox of self-fulfilling prophecies, like the Pygmalion effect. So what then?

Self-fulfilling beliefs don't mean there's no external reality, they just mean your mind (and thus your beliefs) are part of reality, and therefore capable of influencing it. If they weren't, naturally you would be unable to act on them in any case. The correct belief is, of course, "if someone believes X, X will occur. If someone believes Y, Y will occur."

EDIT: The last sentence, which is slightly tangential to the rest, has been moved (on the theory that it was attracting downvotes) to increase the signal-to-noise ratio. It still exists in the comment below, if you wish to downvote it.