We study magnetism and hyperfine structure in Si16 H16 fullerenes endohedrally doped with a transition-metal atom M=Cr, Mn and Fe using density functional theory calculations. In all cases the endohedral fullerene maintains the same magnetic moment as in a free M atom and it is mainly localized on M with partial redistribution on the Si16 H16 cage. However, a comparison between the isotropic hyperfine parameter Aiso of M dopants with that of free M atoms reveals that the electronic wave function at the nucleus of Mn and Fe undergo a significant change both quantitatively and qualitatively as they become doped inside Si16 H16 cage, while for Cr it remains nearly the same. The endohedral doping of Mn and Fe atoms increases the value of the spin density at the nucleus of the M atom, $\rho$s (RM) and correspondingly, the value of Aiso. Analyzing the trend of the core spin-polarization induced by the singly occupied molecular orbitals (SOMOs), it is shown that the increase in the value of Aiso for Mn and Fe atoms comes directly from an increase in the contribution of SOMOs to $\rho$s (RM) and indirectly from hybridization of 3d orbitals of the dopants with the s - and p -like orbitals of the cage. The latter results in a decrease in the spin-polarization of the core 2s and 3s orbitals so that they less effectively contribute to $\rho$s (RM). These results are likely to have wider applicability and could offer a way to identify endohedral doping. {\textcopyright} 2009 The American Physical Society.