We analyse the multi-wavelength properties of about 7500 radio (3-GHz) selected sources in the COSMOS field to investigate the impact of AGN activity on the integrated properties of their hosts. Two main classes of AGN are identified: radiatively- efficient AGN, by combining X-ray, mid-IR diagnostics and SED decomposition, and radiatively-inefficient AGN, that show up only in radio. Interestingly, we find significantly distinct galaxy properties for the two AGN classes, as a function of redshift. At z<1.5, radiatively-inefficient AGN are typically found in more massive and less star-forming galaxies than radiatively-efficient AGN, while at higher redshift we observe a possible reversal of their stellar mass distributions. We interpret these trends in the context of the anti-hierarchical growth of AGN host galaxies, with a particular focus on the role of AGN feedback over cosmic time in radio-selected samples.