In this answer we will stick totally to the Copenhagen interpretation of quantum mechanics.

1. 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.
2. 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.
3. The system has never been in another state (since it did not exist before it was measured).
4. 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.

In this answer we will stick totally to the Copenhagen interpretation of quantum mechanics.

1. 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.
2. 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.
3. The system has never been in another state (since it did not exist before it was measured).
4. 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.