What is an example of LOCC measurement which is not separable?

There isn't: Any LOCC measurement is also a separable measurement. This is easy to see: Alice's first measurement has POVM elements $A_{i_1}otimes I$. Alice then communicates her outcome $i_1$ to Bob. Bob's subsequent measurement has elements $Iotimes A^{i_1}_{i_2}$, where $i_2$ enumerates Bob's outcomes, and $A^{i_1}$ indicates that Bob's POVM can depend on Alice's outcome. The total POVM of both has then elements $$ N_{i_1,i_2}=A_{i_1}otimes B^{i_1}_{i_2} , $$ which is a separable POVM with double index ${i_1,i_2}$. Clearly, this can be iterated to an arbitrary number of rounds, and generalized to an arbitrary number of parties, and will always have POVM elements of the form $N_i=A_iotimes B_iotimes cdots$.

Conversely, not every separable POVM can be written as a LOCC POVM. A counterexample is given in Bennett et al., Quantum Nonlocality without Entanglement, Phys. Rev. A. 59, 1070 (1999).