The effect of B and P dopants on the band structure of Si nanowires is studied using electronic structure calculations based on density functional theory. At low concentrations a dispersionless band is formed, clearly distinguishable from the valence and conduction bands. Although this band is evidently induced by the dopant impurity, it turns out to have purely Si character. These results can be rigorously analyzed in the framework of effective mass theory. In the process we resolve some common misconceptions about the physics of hydrogenic shallow impurities, which can be more clearly elucidated in the case of nanowires than would be possible for bulk Si. We also show the importance of correctly describing the effect of dielectric confinement, which is not included in traditional electronic structure calculations, by comparing the obtained results with those of G0W0 calculations.