>>11766421>>11766426 and >>11767781 are correct but not really relevant because they're missing the point."Observing" a quantum state means you apply a POVM to it. The POVM acts on the entire Hilbert space. There's no notion of "only observing one of the particles", hence there's no notion of "both parties observing their own particles." The POVMs that you and the other person perform are one and the same. There's no contradiction or paradox.
You might ask, well what if Alice measures in a different basis from Bob? The only way you can actually do this is by performing some sort of unitary operation right before measuring. Unitaries are just time evolution generated by a Hamiltonian: it takes nonzero time to change basis. So either Bob measures before she changes her local basis, or he measures in the middle of Alice's unitary, or he waits until after she finishes rotating the state. Regardless of how which procedure they go with, the POVMs they physically apply end up being the same. Certainly they will get different results in every case, because Alice has literally changed the state. But that has nothing to do with whether or not they observe the state "at the same time" or not.
