To depict the impact of surface/interface effects and nonlocal effects on Shear Horizontal (SH) waves in Magneto-Electro-Elastic (MEE) coupled nano-structures, a theoretical model is established, motivated by the dual variable and position (DVP) method. The physical mechanism of the material at the nanoscale is distinctive from that of the counterpart material at the macroscale. The above-referenced phenomenon can be mainly attributed to surface effects and nonlocal effects. With the aid of the DVP method, the phase velocity equation concerning surface effects and nonlocal effects is achieved, which shows stable computation in the high-frequency region. After numerical validation, systematic investigations are carried out to illustrate the size-dependent properties of SH waves. It is revealed that SH₀ waves, i.e. the fundamental mode of the SH wave, are dispersive, with their phase velocity controlled by the nanoplate thickness if surface/interface effects and nonlocal effects are considered, which is totally different from macro-structures. Additionally, the surface/interface effect increases the structural stiffness so that it enhances the phase velocity, while the non-local constant has the opposite effect. This paper establishes a theoretical model and comes up with some qualitative conclusions that pave the way for the design of the structure of MEE nano-devices.