Could you give me an example of a measurement which is separable but not LOCC (Local Operations Classical Communication)? Given an ensable of states $\rho^{N}$, a separable measurement on it is a POVM $\lbrace N_i \rbrace$ where the effects $N_i$ are all of the form $N_i = A_i^{1} \otimes A_i^{2} \otimes \dots \otimes A_i^{N}$. So they are a separable product of effects acting on each state $\rho$ in $\rho^{N}$.
Note: Cross-posted on Physics SE.