John Bell's idea was to establish rigorous limits on the kinds of statistical correlations that could possibly exist between spatially separate events under the assumption of determinism and what might be called local realism. At first he thought he had succeeded, but it was soon pointed out that his derivation implicitly assumed one other crucial ingredient, namely, the possibility of free choice. To see why this is necessary, notice that any two spatially separate events share a common causal past, consisting of the intersection of their past light cones. This implies that we can never categorically rule out some kind of "pre-arranged" correlation between spacelike-separated events - at least not unless we can introduce information that is guaranteed to be causally independent of prior events. The appearance of such "new events" whose information content is at least partially independent of their causal past, constitutes a free choice. If no free choice is ever possible, then (as Bell acknowledged) the Bell inequalities do not apply.
In summary, Bell showed that quantum mechanics is incompatible with a quite peculiar pair of assumptions, the first being that the future behavior of some particles (i.e., the "entangled" pairs) involved in the experiment is mutually conditioned and coordinated in advance, and the second being that such advance coordination is in principle impossible for other particles involved in the experiment (e.g., the measuring apparatus). These are not quite each others' logical negations, but close to it. One is tempted to suggest that the mention of quantum mechanics is almost superfluous, because Bell's result essentially amounts to a proof that the assumption of a strictly deterministic universe is incompatible with the assumption of a strictly non-deterministic universe. He proved, assuming the predictions of quantum mechanics are valid (which the experimental evidence strongly supports), that not all events can be strictly consequences of their causal pasts, and in order to carry out this proof he found it necessary to introduce the assumption that not all events are strictly consequences of their causal pasts!
In the paper "Atomic-Cascade Photons, Quantum Mechanical Non-Locality", Bell listed three possible positions that he thought could be taken with respect to the Aspect experiments. (Actually he lists four, the fourth being "Just ignore it".) These alternatives are
Regarding the third possibility, Bell wrote:
...if our measurements are not independently variable as we supposed...even if chosen by apparently free-willed physicists... then Einstein local causality can survive. But apparently separate parts of the world become deeply entangled, and our apparent free will is entangled with them.
The third possibility clearly shows that Bell understood the necessity of assuming free acausal events for his derivation, but since this amounts to assuming precisely that which he was trying to prove, we must acknowledge that the significance of Bell's inequalities is less clear than many people originally believed. In effect, after clarifying the lack of significance of von Neumann's "no hidden variables proof" due to its assumption of what it meant to prove, Bell proceeded to repeat the mistake, albeit in a more subtle way. Perhaps Bell's most perspicacious remark was (in reference to Von Neumann's proof) that the only thing proved by impossibility proofs is the author's lack of imagination.
This all just illustrates that it's extremely difficult to think clearly about causation, and the reasons for this can be traced back to the Aristotelian distinction between natural and violent motion. Natural motion consisted of the motions of non-living objects, such as the motions of celestial objects, the natural flows of water and wind, etc. These are the kinds of motion that people (like Bell) apparently have in mind when they think of determinism. Following the ancients, people tend to instinctively exempt "violent motions", i.e., motions resulting from acts of living volition, when considering determinism. It's psychologically very difficult for us to avoid bifurcating the world into inanimate objects that obey strict laws of causality, and animate objects (like ourselves) that do not. This dichotomy was historically appealing, but it always left the nagging question of how or why we (and our constituent atoms) manage to evade the iron hand of determinism that governs everything else. This view affects our conception of science by suggesting to us that the experimenter is not himself part of nature, and is exempt from whatever determinism is postulated for the system being studied. Thus we imagine that we can "test" whether the universe is behaving deterministically by turning some dials and seeing how the universe responds, overlooking the fact that we and the dials are also part of the universe.
This immediately introduces "the measurement problem", i.e., where do we draw the boundaries between separate phenomena? What is an observation? How do we distinguish "nature" from "violence", and is this distinction even warranted? It's worth noting that when people say they're talking about a deterministic world, they're almost always not. What they're usually talking about is a deterministic sub-set of the world that can be subjected to freely chosen inputs from a non-deterministic "exterior". But just as with the measurement problem in quantum mechanics, when we think we've figured out the constraints on how a deterministic test apparatus can behave in response to arbitrary inputs, someone says "but isn't the whole lab a deterministic system?", and then the whole building, and so on. At what point does "the collapse of determinism" occur, so that we can introduce free inputs to test the system? Just as the infinite regress of the measurement problem in quantum mechanics leads us into bewilderment, so too does the infinite regress of determinism.
The other loop-hole that can never be closed is what Bell called "correlation by post-arrangement" or "backwards causality". I'd prefer to say that the system may violate the assumption of strong temporal asymmetry, but the point is the same. Clearly the causal pasts of the spacelike separated arms of an EPR experiment overlap, so all the objects involved share a common causal past. Therefore, without something to "block off" this region of common past from the emission and absorption events in the EPR experiment, we're not justified in asserting causal independence, which is required for Bell's derivation. The usual and, as far as I know, only way of blocking off the causal past is by injecting some "other" influence, i.e., an influence other than the deterministic effects propagating from the causal past. This "other" may be true randomness, free will, or some other concept of "free occurrence". In any case, Bell's derivation requires us to assert that each measurement is a "free" action, independent of the causal past, which is inconsistent with even the most limited construal of determinism.
There is a fascinating parallel between the ancient concepts of natural and violent motion and the modern quantum mechanical concepts of the linear evolution of the wave function and the collapse of the wave function. These modern concepts are sometimes termed U, for unitary evolution of the quantum mechanical state vector, and R, for reduction of the state vector onto a particular basis of measurement or observation. One could argue that the U process corresponds closely with Aristotle's natural (inanimate) evolution, while the R process represents Aristotle's violent evolution, triggered by some living act. As always, we face the question of whether this is an accurate or meaningful bifurcation of events. Today there are several "non-collapse" interpretations of quantum mechanics, including the famous "many worlds" interpretation of Everett and DeWit. However, to date, none of these interpretations has succeeded in giving a completely satisfactory account of quantum mechanical processes, so we are not yet able to dispense with Aristotle's distinction between natural and violent motion.