One diagnosis of Bell’s theorem is that its premise of Outcome Independence is unreasonably strong, as it postulates one common screener system that purports to explain all the correlations involved. This poses a challenge of constructing a model for quantum correlations that is local, non-conspiratorial, and has many separate screener systems rather than one common screener system. In particular, the assumptions of such models should not entail Bell’s inequalities. Although we stop short of proving that such models exist (or do not exist), we obtain a few results relating them to models with a common screener system. We prove that if such a model exists, then there exists a local common screener system model for quantum correlations breaking Bell’s inequalities; that model, however, can be conspirational. We also show a way of transforming a model with separate screener systems for the Bell-Aspect correlations that satisfies strong Parameter Independence (PI) and No-Conspiracy (NOCONS) into a somewhat different model for the same correlations in which strong PI is somewhat compromised, but NOCONS and "regular" PI hold, and the elements of the constructed partition act deterministically with respect to measurement results. This means that such models allow the derivation of the Clauser-Horne-Shimony-Holt inequalities.