Consider a bipartite state $\rho$ that is local. By local I mean here that there is a local hidden variable model explaining the correlations resulting from measuring $\rho$. In other words, $\rho$ local means that there are probability distributions $p_\lambda, p_\lambda(a|A),p_\lambda(b|B)$ such that for all projective measurements $\{\Pi^A_a\}_a, \{\Pi^B_b\}_b$ we can write $$\langle \Pi^A_a\otimes\Pi^B_b, \rho\rangle =\int d\lambda \, p_\lambda \, p_\lambda(a|A)p_\lambda(b|B).$$ Nontrivial (as in, possibly entangled) examples of such local state are Werner states $p|\Phi^+\rangle\!\langle\Phi^+|+(1-p)\frac{I}{4}$ for $\frac13<p<\frac12$, as shown e.g. in page 21 of (Brunner et al. 2013). The separability of Werner states is also discussed in What is a separable decomposition for the Werner state?.
For separable states (which are always ensured to be local), the question amounts to the "local hidden variable" in the separable decomposition of the state itself. We know that one can always find a separable decomposition for a bipartite state in $\mathbb{C}^n\otimes \mathbb{C}^m$ that uses at most $n^2 m^2-1$ local states (as discussed e.g. in this physics.SE post), so in this case the associated answer about separability is somewhat already known. At least as far as knowing the cardinality of the required set of values for the local hidden variable.
More generally, is there general technique, result, or reference discussing the "size" of the set of hidden variables required to provide such local explanations for a given quantum state? Of course, there is an immediate issue when asking about size, because a local model might employ a continuous hidden variable. Which makes my question about any result concerning whether a given local state can be described with a discrete, countably infinite, or uncountably infinite local hidden variable.
A related post about the locality of Werner states is How to show that Werner states produce correlations explainable via local hidden variable models?.
