They each get some result and when they compare them they find they
are correlated. Here’s the setup using photons. A pair of photons
whose polarizations are entangled are created by down-conversion in
a crystal and are sent to (possibly distant) polarizers and
detectors.
The detectors record 0 (didn’t pass the polarizer) or 1 (did pass
the polarizer). Alice and Bob keep records of the 0’s and 1’s and
the angle settings in order so that later when they bring their
records together they can calculate the correlation for each angle
setting. For a Bell experiment, they do this for different runs with
their polarizers set at angles 22.5deg and 45deg apart.
Note that in relativity there is no invariant meaning to “at the
exact simultaneous time” at different places. They can be at the
same time in one reference frame, but then they are not at exactly
the same time in a different, moving, reference frame. The
experiment only requires that the measurement events be space-like
separate, i.e. no signal can travel between Alice and Bob so that
the polarizer setting chosen by Alice influences the photon at Bob’s
polarizer and vice versa. Bell’s theorem is that under the
assumption of no-signaling between Alice and Bob a certain
combination of the correlations must always be less than 2. Alain
Aspect (and the other two Nobel recipients this year, Zeilinger, and
Clauser) performed experimental tests of Bell’s theorem and showed
it was violated over a certain range of angles.
The measurements are not made at zero relative angle, so measuring
pass or didn't-pass is not the same at each detector. Rather they
are related probabilistically as shown.
Brent