@Peregrine Not sure what you mean by "publicly accessible" and "empirically". These experiments are done by employing powerful computers, sophisticated optical devices, and millions of data in order to test the correctness of the predicitions of quantum mechanics as accurately as possible. Therefore, it is not easy to see the evidence by simply looking at the data.
What do the researchers do?
(1) They produce pairs of particles , photons in particular, from a single source such as the decay of an atom or some other method. Since the photons come from a single source - the atom in this case-they form an entagnled pair in the EPR sense. I.e. the two photons are described by a single wave function, and from this point of view they are entangled.
(2) They send these pairs of photons to locations, L1 and L2 say, in opposite directions, many miles away from the lab where they have been created.
(3) There, the researchers use aparattus called polarisers to measure the spin of the photons in specified directions, which can also be random. So for each pair of photons created in the lab they have the spin measurements $S_1$ and $S_2$ in locations L1 and L2 respectively.
(4) From a very large number of such spin measurements the researchers calculate statistical correllations of photon spins along those random directions.
(5) These correlations are then used to calculate the terms in Bells inequality, which has a very well known form. If the correlations of the spin measurements satisfy Bell's inequality, it would mean that something is missing from quantum mechanics the way it stands. More precisely, it would support the existence of hidden variables and locality. If on the other hand Bell's inequality is violated, then hidden variables cannot exist and quantum mechanical predictions are correct; i.e. quantum mechanics is a complete theory with non-local influences. Expeiments done todate violate Bell's inequality which give suport to quantum mechanical predicition.
The radomness in the direction of photon-spin measurements, is a way of testing the effect of entaglement and non-locality under unknown and random conditions, and yet quantum mechanics comes out clean in its predictions. I hope this helps you estimate the significance of thse experiments.