We present numerical simulations that include 1D Eulerian multigroup radiation-hydrodynamics, 1D non-local thermodynamic equilibrium (non-LTE) radiative transfer, and 2D polarized radiative transfer for superluminous interacting supernovae (SNe). Our reference model is a ∼10 M⊙ inner shell with 1051 erg ramming into an ∼3 M⊙ cold outer shell (the circumstellar medium, or CSM) that extends from 1015 to 2 × 1016 cm and moves at 100 km s−1. We discuss the light-curve evolution, which cannot be captured adequately with a grey approach. In this type of interactions, the shock-crossing time through the optically thick CSM is much longer than the photon diffusion time. Radiation is thus continuously leaking from the shock through the CSM. This configuration is distinct from the shell-shocked model. Our spectra redden with time, with a peak distribution in the near-UV during the first month gradually shifting to the optical range over the following year. Initially, Balmer lines exhibit a narrow line core and the broad line wings that are characteristic of electron scattering in the SNe IIn atmospheres (CSM). At later times, they also exhibit a broad blue-shifted component which arises from the cold dense shell. Our model results are broadly consistent with the bolometric light curve and spectral evolution observed for SN 2010jl. Invoking a prolate pole-to-equator density ratio in the CSM, we can also reproduce the ∼2 per cent continuum polarization, and line depolarization, observed in SN 2010jl. By varying the inner shell kinetic energy and the mass and extent of the outer shell, a large range of peak luminosities and durations, broadly compatible with superluminous SNe IIn like 2010jl or 2006gy, can be produced.