The “message” does not have any local effect on reality - when you measure your particle, you have no way of figuring out if its partner was already measured elsewhere. The effect it does have is on the global state, maintaining the correlation that was encoded from the start.

If you write up the density matrix for the system before and after measurement of one of the particles, you can see that while the density matrix changes, it does not change in a way one can measure.

What I will concede is that before the first measurement the global state is |00>+|11>, afterwards it is |00> or |11>. This projection appears to happen instantaneously, no matter the distance, which is indeed faster than light.

But calling the wave function collapse a signal or a message or a transfer of information is misleading, I would say. In your example, we know that the initial state is |00>+|11>, and that the result of the first measurement is then, say, 1. Then no further information is required to know that the other measurement will result in 1. No messages required, no hidden variables, simply the process of elimination.

I would like to say that this is indeed a confusing subject, but that the math is clear, and that I am arguing what is my impression of the mainstream view in the field.