Spherical Janus colloids, particles with different surface properties on their two hemispheres, are generally heterogeneous in permittivity. This dielectric heterogeneity may affect their behavior in electrolytes and external electric fields, but is typically not taken into account in computational studies. We apply the iterative dielectric solver developed by Barros and Luijten [Phys. Rev. Lett., 2014, 113, 017801] in combination with preconditioning techniques that can accurately and efficiently compute the polarization of dielectrically anisotropic particles. Employing this approach, we systematically study the ion distribution around neutral and charged Janus particles with various permittivities, immersed in symmetric and asymmetric electrolytes. We demonstrate that neutral Janus colloids may carry a nonzero dipole moment in asymmetric salts. For charged Janus colloids, dielectric effects can substantially influence the electric double layer. These findings also have implications for other dielectrically anisotropic entities, such as proteins.