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).
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