In this answer we will stick totally to the Copenhagen interpretation of quantum mechanics.
- Before the system is measured, there is nothing physical (a consequence of realism not playing a part under this interpretation). There is only our mathematical, non-physical description of the possible measurement outcome called the wavefunction.
- On measurement the system is made to exist in a physical way. And is created in one of the eigenstates under which we are measuring.
- The system has never been in another state (since it did not exist before it was measured).
- Thus there is no need for one 'particle' to tell the other what state it is in because there is only one state which is physical.