  We investigate the antiferromagnetic insulating state of the recently
discovered double perovskites Sr$_2$XOsO$_6$ (X$=$Sc, Mg) by using ab-initio
calculations (based on Density Functional Theory and Dynamical Mean-Field
Theory) to elucidate the interplay between electronic correlations and
spin-orbit coupling. The structural details of Sr$_2$XOsO$_6$ (X$=$Sc, Mg)
induce band narrowing effects which enhance local electronic correlations. The
half-filled $5d^3$ orbitals of Os in Sr$_2$ScOsO$_6$ fall into a magnetically
ordered correlated regime, which is slightly affected and reduced by the
spin-orbit coupling. The electronic configuration $5d^2$ of Os in
Sr$_2$MgOsO$_6$ responses differently to electronic correlations promoting a
less localized state than Sr$_2$ScOsO$_6$ at the same strength of electronic
interactions. We find that the inclusion of spin-orbit coupling drives
Sr$_2$MgOsO$_6$ toward insulating behaviour and promotes a large tendency in
formation of orbital magnetization antiparallel to the spin moment. The
formation of the AFM state is linked to the evidence of correlated Hubbard
bands in the paramagnetic solution of Sr$_2$XOsO$_6$ (X$=$Sc, Mg).
