In this paper we study the electrostatic properties of "Janus" spheres with unequal charge densities on both hemispheres. We introduce a method to compare primitive-model Monte Carlo simulations of the ionic double layer with predictions of (mean-field) nonlinear Poisson-Boltzmann theory. We also derive practical Derjaguin Landau Verwey Overbeek (DLVO)-like expressions that describe the Janus-particle pair interactions by mean-field theory. Using a large set of parameters, we are able to probe the range of validity of the Poisson-Boltzmann approximation, and thus of DLVO-like theories, for such particles. For homogeneously charged spheres this range corresponds well to the range that was predicted by field-theoretical studies of homogeneously charged flat surfaces. Moreover, we find similar ranges for colloids with a Janus-type charge distribution. The techniques and parameters we introduce show promise for future studies of an even wider class of charged-patterned particles.