PeterDonis said:
No. There is no ##\lambda_1## and ##\lambda_2##. There is just ##\lambda##. The whole point of ##\lambda##, as I have already said, is that it denotes hidden variables that are in the causal past of both measurements and therefore can be taken to be common to both measurements. There is no such thing as some pieces of ##\lambda## traveling to measurement ##A## and other, different pieces of ##\lambda## traveling to measurement ##B##. There is just ##\lambda##. That is what appears in Bell's math.
##\lambda## can be any number of "hidden variables". The strong point of Bell's derivation of his inequalities is the generality.
The problem is the very confusing language involved, which is due to EPR. Einstein himself was not happy with the EPR paper, and he wrote a much more concise paper in 1948. The point he was uneasy with was what he called "inseparability", i.e., the fact that with entanglement in QT there are systems, where far distant parts (or more precisely where observations on far distant parts) of a quantum system are strongly correlated, but the measured observables showing these correlations are maximally indetermined. The most simple example, going back to Bohm, is the entanglement between spin components for two particles from a decay of a spin-0 particle, which are in a spin-singlet state. If nothing disturbes this particles you can wait until they are arbitrarily far away from each other, and although the spin-##z## components are maximally indetermined (i.e., each of the particles is completely unpolarized) there's a strong anti-correlation in the outcomes of measurements, i.e., if particle 1 is found with ##\sigma_z=+1/2##, the other is found with ##\sigma_z=-1/2## and vice versa. Measuring certain combinations of spin components (which is of course possible only on ensembles of equally prepared particles) demonstrates the violation of Bell's inequalities.
In Bell's "local, realistic" hidden-variable theories you have the assumptions
(a) all observables always take determined values. The probabilistic description is necessary because of our ignorance about the hidden variable(s), ##\lambda##. There is a probability distribution ##P(\lambda)## for these variables. This is "realism".
(b) local observables of far distant parts are independent, i.e., the outcome of the measurement of an observable ##A## at one place and of an observable ##B## at another far distant place are independent of each other, i.e., they are determined by the setup of the measurement devices, ##a## and ##b## (e.g., in the above mentioned Bohm spin-1/2 example the choice of the measured spin components on particles at ##A## and ##B##) and the hidden variables only: ##A=A(a,\lambda)## and ##B=B(b,\lambda)##, i.e., ##B## doesn't depend on ##a## and ##A## doesn't depend on ##B##. Both, however, depend on ##\lambda##. That's what somewhat unfortunately is called "locality", although it is in fact "separability".
The mathemically clear description of a "local, realistic HV theory" simply is that the correlation function is given by
$$\langle A B \rangle = \int \mathrm{d} \lambda A(a,\lambda) B(b,\lambda) P(\lambda).$$
As far as I know the literature the realization of "inseparability" is usually considered fulfilled with certainty precisely by the realization of the local measurments on the far distant parts of the system in such a way that the "measurement events" are space-like separated, i.e., that there cannot be any causal influences of the measurement on B by the measurement on A and vice versa. This of course hinges on the fufillment of relativistic causality constraints, and indeed all relativistic theories are by construction fulfill this requirement.
This is realized in both the classical as well as the quantum realm by the paradigm of locality, i.e., the field concept. In classical relativistic physics this means that interactions between far distant parts (e.g., particles) of a system is due to local field actions. E.g., a particle "feels" the electromagnetic interaction with another particle due to the field due to this other particle's electromagnetic field at the spacetime point, ##q##, of the former particle, ##\vec{K}^{\mu}=q F^{\mu \nu}(x)##, and ##F^{\mu \nu}## is the retarded solution of the Maxwell equations with the other particle's electric-charge four-current as the source.
For QT this classical paradigm of "local" field interactions is realized by formulating it as local relativistic QFT, i.e., the local observable operators (densities of energy, momentum, angular momentum, charge-current densities) are commuting at space-like distances of their arguments, which includes the Hamilton density of the system, which ensures that there cannot be any causal influences between space-like separated measurement events.
In this sense local relativistic QFTs are indeed "local" but, of course, not "realistic", and that's why at least within this interpretation of QFT (statistical/ensemble interpretation), where there's nothing else than the probabilities, including the strong correlations described by entanglement, predicted by QFT, the quibble of EPR concerning relativistic causality is resolved.
