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If disorder-induced Anderson localized states have been observed experimentally in optics, their study remains challenging leaving a number of open questions unsolved. Among them, the impact on Anderson localization of non-Hermiticity, optical gain and loss, and more generally, nonlinearities has been the subject of numerous theoretical debates, without yet any conclusive experimental demonstration. Indeed, in systems where localized modes have reasonable spatial extension to be observed and investigated, their mutual interaction and coupling to the sample boundaries make it extremely difficult to isolate them spectrally and investigate them alone. Recently, we successfully exhibited localized lasing modes individually in an active disordered medium, using pump-shaping optimization technique. However, a one-to-one identification of the lasing modes with the eigenmodes of the passive system was not possible, as the impact of non-Hermiticity and nonlinear gain on these localized states was unknown. Here, we apply the pump-shaping method to fully control the non-Hermiticity of an active scattering medium. Direct imaging of the light distribution within the random laser allows us to demonstrate unequivocally that the localized lasing modes are indeed the modes of the passive system. This opens the way to investigate the robustness of localized states in the presence of nonlinear gain and nonlinear modal interactions. We show that, surprisingly, gain saturation and mode competition for gain does not affect the spatial distribution of the modes.