It's actually a bit unlikely for the following reason:
Dark matter is observed to not 'clump' as effectively as ordinary matter. This is due to the fact that ordinary matter feels drag due to interactions with the intergalactic dust and loses angular momentum to radiation, causing it to fall inward toward the galactic core, and to fall toward a central disk that is prevented from further collapse due to conservation of angular momentum.
Dark matter, however, is observed to maintain an approximately spherical distribution, rather than mirroring the external disk, and is thus expected to be immune to the interactions that caused the visible matter in, say, the milky way to flatten and fall a bit inward. since the majority of the matter in most galaxies is dark matter, if the dark matter interacted with itself in such a way that "dark radiation" was produced, you'd expect qualitatively similar behaviour to the ordinary matter.
That said, I'm sure you could construct models that worked.