Using first principle calculations, the problem of scaling magnetic properties in nanoparticles is addressed. To this aim, the local electronic structure is characterized in cobalt quasi-spherical magnetic nanoparticles, in a large size range, from 0.5 to 2 nm of diameter. First, specific patterns of the magnitude of local spin magnetic moments are evidenced depending on the shape and the size of the nanoparticles. Then, effects of local structural environment (atomic coordination, structural deformations, finite size effects, shape changes) are unravelled. In small icosahedral nanoparticles, the local spin magnetic moment is found to decrease from the surface to the center. General rules driving charge transfers are observed whereby donor atomic sites are exclusively subsurface atoms and more unexpected vertex surface atomic sites. The variation of the magnetic moment is driven by the coupling between cluster microstructure and complex hybridization effects. In larger truncated octahedral clusters, whereas some properties are still found (quasi-zero inter-atomic site charge transfer, the reversal from anti-ferromagnetic to ferromagnetic coupling of electronic sp states with d ones at vertex atomic sites), other vary such as the behavior of the local spin magnetic moment which now presents weak oscillations.