For a classical formulation of the EPR paradox, two particles are produced, with total momentum zero and separated by a long distance. So say we measure the momentum of one particle first, and measure it with absolute certainty. Heisenberg uncertainty principle means that we can not know both position and momentum. But as we know the momentum of one particle, we know the momentum of the other. So we can not know the position of the second particle. So what happens when we try to measure the position of the second particle? We should not be able to do it, poisition of the particle should be infinitely uncertain. But what does it meson physically? Doesn’t it mean that all position measurements should fail? Then would it mean that information is somehow passed: you try to measure electron position and fail, thus you know its momentum has been previously measured. I think the answer is in the details of how the measurement is actually made, but I am failing to visualize it.
Well, how do you know where the electron, of which you are about to measure the momentum, is located in the first place? By searching for an electron. Searching for an electron, I think, is not the same as measuring the position of an electron.
Once you found an electron and make a measurement of its momentum, you never (in practice) make an exact measurement, because this requires a photon wavepacket that's infinitely small. So there can't be an infinite uncertainty of the electron's position (theoretically it's possible, but in practice never ever).
So what does this mean for measuring the position of one of the two electrons (the question didn't have to involve entanglement)? I think it would be hard to find your original electron(s) back, because of the big uncertainty (though not infinite uncertain). But you can estimate the volume of space where it (they) can be in, and send a photon with an associated wave packet that has about the same size (volume, in 3-d) as that volume. It will, of course, be possible you measure the position of another electron that happens to be also in that volume. But because they are indistinguishable particles, you won't be able to tell if you've measured the position of your original electron. I shouldn't know how to label your original electron. Maybe you should paint it black...