We don't know. More specifically, the answer to your question is interpretation-dependent.
In objective-collapse interpretations, for example, you do have an explicit collapse of the wavefunction of the first particle to be measured, followed by an instantaneous (?) collapse of the entangled partner, with all the special-relativity problems brought about by the unspecified temporal ordering of the two measurements if their separation is spacelike. To try and make sense of that, you need to allow the particles to 'communicate' their states superluminally; that said, the randomness of the collapse 'protects' the superluminal communication and makes it inaccessible to us.
If you don't like it, then well, that's just how that interpretation looks like, so you can turn to the epistemological comforts of, say, the Many-Worlds Interpretation, which will just tell you that if you measure one of the parties of an entangled state, all you do is become a part of what is now a tripartite entangled state, and the information transfer... what information transfer?
Ultimately, Bell's theorem (and the subsequent observations of Bell-inequality violations in experiment) tells us that, if nature is 'real' at some underlying level, then that reality needs to be nonlocal in some sense. While we have good grasps on certain bounds that that nonlocality needs to obey (like the no-communication theorem) we still don't understand the nature of that nonlocality as well as we would like. And frankly, anybody that tells you that we understand it is either fooling you or fooling themselves.