Manipulating light at the nanoscale by means of dielectric nanoantennas recently received renewed attention thanks to the development of key enabling fabrication tools in semiconductor technology, combined with the extremely low losses exhibited by dielectrics in the optical regime. Nanostructures based on III-V type semiconductors, characterized by an intrinsic broken symmetry down to a single elementary cell, has already demonstrated remarkable nonlinear conversion efficiencies at scales well below the operating wavelength. In this Letter, we thoroughly investigate the emission properties of second-harmonic generation (SHG) in AlGaAs monolithic nanoantennas. Our findings point toward the pivotal role of volume susceptibility in SHG, further unraveling the physics behind the nonlinear processes in these systems. The extremely high SHG efficiency attained, together with the control over the polarized emission in these nanoantennas, constitute key ingredients for the development of tunable nonlinear metasurfaces.