You seem to think of entanglement as a property of individual particles. That is not true:
Let $\mathcal{H}$ be the Hilbert space of states of a single particle (electron, photon, whatever, doesn't matter). Then the space of states of two particles is given by the tensor product $\mathcal{H}\otimes\mathcal{H}$, and the space of states of $N$ particles is given by $\bigotimes_{i = 1}^N \mathcal{H}$. A state $\lvert \psi \rangle \in \bigotimes_{i = 1}^N \mathcal{H}$ is called entangled if it cannot be written as the product of individual states, i.e. if there are no $\lvert \psi_i \rangle \in \mathcal{H}$ such that
$$ \lvert \psi \rangle = \lvert \psi_1 \rangle \otimes \dots \otimes \lvert \psi_N \rangle $$
And that's it. This does not tell you anything about the state other than that there are no definite states of the single particles from which the entangled state would arise. In particular, there is no property of the individual particles that would imply entanglement, since entanglement precisely means that there are no well defined individual particle states in an entangled state.
There is also no communication happening in the naive sense. For a two-particle entangled state, $\lvert \psi \rangle = \lvert \psi_1^A \rangle \otimes \lvert \psi_2^A \rangle + \lvert \psi_1^B \rangle \otimes \lvert \psi_2^B \rangle$, a measurement of a single particle simply "collapses" the state into either $\lvert \psi_1^A \rangle \otimes \lvert \psi_2^A \rangle$ or $\lvert \psi_1^B \rangle \otimes \lvert \psi_2^B \rangle$. In a QM interpretation with collapse, this seems to require some sort of communication (though you'd be hard-pressed to even say how that "signal" could ever be transferred), but in non-collapse interpretations (e.g. Many Worlds), there is no need for communication in the naive sense, since the state simply evolves happily ever after, and we are now just also entangled with it.
But whatever interpretation you may or may not adhere to, there is nothing in the math requiring anything like an actual signal of any kind to be sent through the electric field or whatever.
