I think you would have a shared private piece of correlated information between each other that wasn't determined until you made the measurement (though maybe no joint reference frame to say who made it first), but you can't choose what it was (communicate with each other).

The universe either had to break the light barrier to make the measurements correlated (predetermining the outcome isn't generally possible because you could choose how to make the measurement based on another quantum measurement from something outside of the other participant's then-current light cone), or make the same choice through superdeterminism (the other measurement and all others were predetermined too and exact simulation of entire future universe's measurement decisions was shared between every particle when they were within some distance at big bang or something). But even though the universe broke the light barrier, you yourself aren't able to use it for communication.

In the many-worlds interpretation you've both branched into the same branch of the multiverse, but couldn't choose which branch. You do have private knowledge of which branch and the consequences of that, assuming you both followed the agreed on procedure.

I think you can use what you are describing in a series of correlated measurements to set up a provably secure one-time-pad, and then do secure classical communication with it. But you don't communicate the actual bits of the pad, you just both get correlated ones.

You didn't gain any information after parting - you'd "know" just as much if your compatriot on Earth had given you a sealed envelope that said "Russia" or "America" inside.

You can take actions that will later turn out to be correlated with each other, but you can only find that out once you meet up again, bounded by the speed of light.