Current research develops a comprehensive wave propagation analysis of a magneto-electro-thermo-elastic (METE) nano-beam (NB) resting on the visco-elastic medium. To model the size dependency effects, modified couple stress (MCS) and Eringen's nonlocal (ENL) theories are employed to analyze and describe the wave propagation behaviors for those nano-beams. These theories were the most used in the literature due to the inclusion of one additional size-dependent length scale parameter. Those theories are compared side by side in this investigation and their impacts/differences on the wave propagation of specific materials are explored. Sinusoidal shear deformation beam model with Hamilton's principle is adopted to develop the governing equations of motion. Then, an analytical solution is executed to extract numerical results for transverse wave propagation in both elastic and METE configurations of the nano-beam. The effects of size-dependent length scale of both theories, thickness of NB, Winkler-Pasternak coefficients, thermal gradient, and magnetic potential and external electric voltage are illustrated and discussed in details. It is concluded that wave frequency decreases with increment of nonlocal parameter for ENL model. On the other hand, a stiffening effect takes place for the wave frequency when the MCS model is considered. Hence, the results indicate that there is a significant difference between ENL and MCS theories in the estimation of the behavior of wave propagation in small-scale structures. One of the main results of this investigation indicates that the MCS theory has similar nonlocal effects as the Eringen's theory for specific conditions.