The elastoplastic behavior of sodium silicate glasses is studied at different scales as a function of composition and pressure, with the help of quasistatic atomistic simulations. The samples are first compressed and then sheared at constant pressure to calculate yield strength and permanent plastic deformations. Changes occurring in the global response are then compared to the analysis of local plastic rearrangements and strain heterogeneities. It is shown that the plastic response results from the succession of well-identified localized irreversible deformations occurring in a nanometer-size area. The size and the number of these local rearrangements, as well as the amount of internal deviatoric and volumetric plastic deformation, are sensitive to the composition and to the pressure. In the early stages of the deformation, plastic rearrangements are driven by sodium mobility. Consequently, the elastic yield strength decreases when the sodium content increases, and the same when pressure increases. Finally, good correlation was found between global and local stress-strain relationships, reinforcing again the role of sodium ions as local initiators of the plastic behavior observed at larger scales.