Recent work has extended Bell's theorem by quantifying the amount of communication required to simulate entangled quantum systems with classical information. The general scenario is that a bipartite measurement is given from a set of possibilities and the goal is to find a classical scheme that reproduces exactly the correlations that arise when an actual quantum system is measured. Previous results have shown that, using local hidden variables, a finite amount of communication suffices to simulate the correlations for a Bell state. We extend this in a number of ways. First, we show that, when the communication is merely required to be finite on average, Bell states can be simulated without any local hidden variables. More generally, we show that arbitrary positive operator valued measurements on systems of n Bell states can be simulated with O(n2n) bits of communication on average (again, without local hidden variables). On the other hand, when the communication is required to be absolutely bounded, we show that a finite number of bits of local hidden variables is insufficient to simulate a Bell state. This latter result is based on an analysis of the nondeterministic communication complexity of the NOT-EQUAL function, which is constant in the quantum model and logarithmic in the classical model.