Common models of blazars and gamma-ray bursts assume that the plasma underlying the observed phenomenology is magnetized to some extent. Within this context, radiative signatures of dissipation of kinetic and conversion of magnetic energy in internal shocks of relativistic magnetized outflows are studied. We model internal shocks as being caused by collisions of homogeneous plasma shells. We compute the flow state after the shell interaction by solving Riemann problems at the contact surface between the colliding shells, and then compute the emission from the resulting shocks. Under the assumption of a constant flow luminosity, we find that there is a clear difference between the models where both shells are weakly magnetized (σ≲ 10−2) and those where, at least, one shell has σ≳ 10−2. We obtain that the radiative efficiency is largest for models in which, regardless of the ordering, one shell is weakly and the other strongly magnetized. Substantial differences between weakly and strongly magnetized shell collisions are observed in the inverse-Compton part of the spectrum, as well as in the optical, X-ray and 1-GeV light curves. We propose a way to distinguish observationally between weakly magnetized and strongly magnetized internal shocks by comparing the maximum frequency of the inverse-Compton part and synchrotron part of the spectrum to the ratio of the inverse-Compton to synchrotron fluence. Finally, our results suggest that low-frequency peaked blazars (LBL) may correspond to barely magnetized flows, while high-frequency peaked blazars (HBL) could correspond to moderately magnetized ones. Indeed, by comparing with actual blazar observations, we conclude that the magnetization of typical blazars is σ≲ 0.01 for the internal shock model to be valid in these sources.