In short, the only information you gain is about the outcome of the other side's measurement. They cannot introduce information into the particle, and thus can't transmit information. The only thing you learn is what you already knew: the other particle had a 50% chance of being in one state or another, with the added fact that it's now correlated to your particle with a 75% (depending on experiment) chance. This is information that didn't exist until that moment, so it couldn't have been sent out and reach you before you measure your particle, which would inform you with 75% certainty how your particle would act and break causality.