In your question, you were also wondering what would happen if both Alice and Bob apply the same Pauli gate on their respective qubit. To complement Mark's answer, I would like to expand on this point. Interestingly, the answer is that nothing happens, the action of their individual gates is equivalent to the identity map! This is a consequence of the more general result that for any maximally entangled pure state $|\psi\rangle$ and any unitary $U$, it holds that
where $U^T$ is just the transpose of $U$. All the Bell states are maximally entangled and the Pauli operators satisfy $\sigma_x^T=\sigma_x$, $\sigma_y^T=-\sigma_y$, $\sigma_z^T=\sigma_z$ so that if both Alice and Bob apply the same Pauli gate, from the above result the state is unchanged (up to a global phase).