We have used atomistic simulations to study the role of electrostatic screening and charge correlation effects in self-assembly processes of ionic surfactants into micelles. Specifically, we employed grand canonical Monte Carlo simulations to investigate the critical micelle concentration (cmc), aggregation number, and micellar shape in the presence of explicit sodium chloride (NaCl). The two systems investigated are cationic dodecyltrimethylammonium chloride (DTAC) and anionic sodium dodecyl sulfate (SDS) surfactants. Our explicit-salt results, obtained from a previously developed potential model with no further adjustment of its parameters, are in good agreement with experimental data for structural and thermodynamic micellar properties. We illustrate the importance of ion correlation effects by comparing these results with a Yukawa-type surfactant model that incorporates electrostatic screening implicitly. While the effect of salt on the cmc is well-reproduced even with the implicit Yukawa model, the aggregate size predictions deviate significantly from experimental observations at low salt concentrations. We attribute this discrepancy to the neglect of ion correlations in the implicit-salt model. At higher salt concentrations, we find reasonable agreement of the Yukawa model with experimental data. The crossover from low to high salt concentrations is reached when the electrostatic screening length becomes comparable to the headgroup size.