Definition: Radiation in this case does not refer to electromagnetic radiation. It refers to any kind of emission of energy, even energy that does not interact with regular matter.
Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with "dark matter radiation" (I'm not talking about the usual "really high wavelength radiation" kind)?
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.
In order to address the question :
Just like dark matter does not interact with electromagnetic radiation, could regular matter not interact with "dark matter radiation" (I'm not talking about the usual "really high wavelength radiation" kind)?
One needs experimental evidence, as @dmckee 's comment suggests.
The other tack is to find a Theory Of Everything which describes experimental data in particle physics and check whether there are predictions for dark energy and dark matter behavior in specific models which will address the interactions of dark matter with the known matter.
The only consistent candidate models are based on string theory, which allows for the quantization of gravity and for the inclusion of the known data as described by the standard model of particle physics.
Various models based on strings have candidate new particles for dark matter and will have coupling constants for computing the interaction of these dark matter candidates with radiation fields that the theory allows. These models have predictions that may be realized in the experiments now running at the LHC at CERN, and if they are confirmed, then the question of interaction of normal matter and dark matter will also be addressed by the model.
Well, I claim the answer is yes! Since the question defines radiation as emission of energy that doesn't interact with normal matter, I say that dark matter emitting dark matter particles qualifies as emission of energy that doesn't interact with normal matter. An clearly it can do that.
Yes, just as you have bosons mediating interactions in the Standard Model sector, you could have dark radiation mediating interactions in the DM sector. Given that more energy is in the form of DM than SM particles, this won't be surprising. Of course, one needs proof for such a species of particles to exist :) A particle physics approach would be to understand the exact nature of the DM particle and its interactions. A cosmological approach would be to see if there are missing relativistic components, as in this paper.
By the way, by radiation, I hope you mean relativistic particles. It is in that sense that radiation is used in the cosmological context.
