[betty boop]

But the photons themselves are in an indeterminate polarization state, until one of them is measured. When the polarization of one of them is measured, it collapses into a well-defined polarization state.... When the other photon is measured, its polarization collapses into a well-defined state, too....

Let me see if I'm following you here, Physicist. When one photon is measured, and collapses into a well-defined polarization state, its "twin" is similarly affected, though it be light years away from the first one. Is the subsequent measurement of the second photon even necessary if all the relevant information was given in the collapse of the first?

[Physicist]

When one photon is measured, and collapses into a well-defined polarization state, its "twin" is similarly affected, though it be light years away from the first one.

I don't want to call that wrong, because that is how it is most commonly portrayed. It would be more accurate to say that there is one state that collapses once, but as that state is not tied to a specific location, the time of its collapse cannot be fixed. You see, the two measurement events most likely have a spacelike separation. The order of the two measurements is therefore frame-dependent, so it's just as reasonable to say that the "second" measurement caused the collapse of the state.

Is the subsequent measurement of the second photon even necessary if all the relevant information was given in the collapse of the first?

I guess I don't understand the question. Necessary to do what? One measurement is enough to collapse the state, but both are necessary to measure the correlation.