In reality there is no way to use quantum entanglement to communicate non-locally and there is a non communication theorem to that effect, which is entirely uncontroversial, see Section II E of this review:
https://arxiv.org/abs/quant-ph/0212023
I could leave it at that but I will explain a bit more assuming that the equations of motion of quantum theory are used to describe what's going on including measurements.
A measurement is an interaction that produces a record of some property of a system. In your proposal there are two systems $A$ and $B$ in the state
$$|\phi\rangle_{AB}=\tfrac{1}{\sqrt{2}}(|0\rangle_A|0\rangle_B+|1\rangle_A|1\rangle_B)$$
where $|0\rangle,|1\rangle$ are the $+1,-1$ eigenstates of $\sigma_z$. The corresponding eigenstates for $\sigma_x$ are $|0\rangle_x=\tfrac{1}{\sqrt{2}}(|0\rangle+|1\rangle),|1\rangle_x=\tfrac{1}{\sqrt{2}}(|0\rangle-|1\rangle)$.
A perfect measurement of an observable $\hat{A}$ with eigenvectors $|j\rangle_S$ of a system $S$ onto a measurement device $M$ in a ready state $|0\rangle_M$ has the effect:
$$|j\rangle|0\rangle_M\to|j\rangle|j\rangle_M$$
and doesn't change any system other than the measuring device and the measured system. The measurement operator would be
$$U_{SM}=\sum_{j=1}^d|j\rangle|j+k\rangle_M\langle j|\langle k|_M$$
where $+$ stands for addition modulo $d$, which is the dimension of the Hilbert spaces of $S$ and $M$.
So if the initial state is
$$|\Psi(0)\rangle=|\phi\rangle_{AB}|0\rangle_{MA_1}|0\rangle_{MA_2}|0\rangle_{MB}$$
where $MA_1,MA_2,MB$ denote the measurement devices used to do the measurements on $A_1,A_2,B$ respectively.
The evolution operators for the $A_1$ measurement is then
$$U_{AMA_{1}}\otimes I_{MA_2}\otimes I_{B}\otimes I_{MB}$$
where $I_{whatever}$ is the identity operator on that system. So the $A_1$ measurement changes $A$ and $MA_{1}$ and nothing else. And you can write down the measurement operators for the other systems and reach similar conclusions for the other measurements. Doing the maths might help you understand the situation better.
It is commonly claimed that measurements have non-local effects even if they can't be used for communication but this isn't an implication of the equations of motion of quantum theory. Using those equations of motion the local flow of information in the Heisenberg picture, see
https://arxiv.org/abs/quant-ph/9906007
https://arxiv.org/abs/1109.6223
