Among the two materials families used in nanophotonics, the fundamental mode for metal nanostructures is electric, while that for dielectric nanostructures is magnetic. Consequently, the optical properties of hybrid dimers that incorporate both materials have prompted significant research. Here, we study the optical response of such hybrid dimers with the coupled electric and magnetic dipoles method, and demonstrate how their electromagnetic interactions can be used to modulate the electric and magnetic responses. The conditions for the complete suppression of the magnetic dipole in the visible range for high refractive index dielectric nanoparticles are described. The electric and magnetic responses can be enhanced or reduced by positioning the dielectric particle in the nodes of the standing wave formed by the metallic particle. This controlled near-field interaction provides a handle on the near- and far-field responses of the system, with possible applications such as sensing and optical switches.