So we measure the particles and find that they are in inverse states. I assume there's a reason for this, but why are we sure that we're not just reading off a random number generator that's set from the same seed?
It's my sacred duty to keep repeating this fact until the whole internet knows. (It's even in the first sentence of the wikipedia page on quantum teleportation!)
If you don't get the classical information on the receiving end, any attempts to read the quantum state will garble the quantum transmission such that it'll be indistinguishable from random noise. Since those classical bits are limited by the speed of light, so is the overall transmission of the state.
I'm not prone to study too deeply because I've learned (apparently) very bad analogies or schemes for entanglement.
The receiver needs the two bits to choose which of four kinds of garbling occurred, so it can correct them with the appropriate post-processing (if necessary).
We aren't. In fact, according to the decoherence / many-world interpretation (and possibly others) that's exactly what happens.
When you "measure" (aka. entangle yourself with) one of the two photons, you end up (find yourself) in one of the infinitely many possible "worlds", in which both photons have a single, definite state (always the opposite of each other, in this setup.) It's as simple as that.
I keep recommending this sequence of blog posts as the best introduction to the topic: