Combining density functional calculations and solutions to the steady-state Boltzmann transport equation, the transport properties of osmium and osmium silicide are studied. The calculations of electrical and thermal resistivity take full account of the ab initio electron-phonon coupling, while the Seebeck coefficient is calculated in the relaxation-time approximation. Our predicted resistivities of Os are in good agreement with experiment at both low and high temperatures, which implies electron-electron scattering is less important than electron-phonon scattering, even for temperatures below 20 K. Intrinsic OsSi is a narrow-gap semiconductor, and our prediction of the band gap confirms the indirect estimation from the measured resistivity. This agreement results from a common cancellation of error in density functional theory with spin-orbit coupling. For the degenerate semiconductor OsSi, we determine the range of carrier concentrations present in experiments, through comparison with transport data. We further predict the power factor and figure of merit for both $n$ and $p$ types, with the dependence on doping concentration and temperature.