This is an interesting question, because the electromagnetic version has been studied extensively and has been controversial. Any arguments about the electromagnetic version should probably carry over, at least qualitatively, to the gravitational case. However, the gravitational version doesn't threaten the equivalence principle, which is presumably why you don't see it discussed as much.
It seems pretty obvious to me that an observer at infinity will see radiation. After all, we detect gravitational waves from inspiraling binary systems, and the fact that this example is radial shouldn't matter. (IIRC one way of stating the criterion for radiation is that the mass quadrupole moment has to have a nonvanishing third time derivative, which is certainly true in this thought experiment.)
For an observer comoving with one body or the other, maybe a little bit more delicacy is required.
In the discussion of the electromagnetic case, one of the fundamental issues is how to define a radiation field. It isn't an unambiguously defined thing, and this is why you can get apparently contradictory answers to the question of whether falling charges radiate -- both "yes" and "no" can be correct, depending on your definition of a radiation field.
Because the system is not a black hole binary, the motion of the two objects is at nonrelativistic speeds, and therefore the wavelength of the gravitational waves is large compared to the size of the system. This means that two observers falling along with the two objects will both be in the near-field region, and will both be essentially observing the same wave. So I guess we have to ask whether there's some unambiguous way to define what is or is not a radiation field, purely locally, in the near-field region. I'm not sure that there is.
It is certainly going to be true that each observer will see their own object accelerate. This is not a violation of the equivalence principle, because the EP only predicts universality of free-fall motion in the case of test particles. In that limit, radiation effects vanish. I think the only issue is whether those observers would describe these anomalous accelerations as being due to a gravitational wave. You could try to resolve this ambiguity by saying, OK, will LIGO detect a gravitational wave in this situation? The problem then is that LIGO is sensitive to all sorts of gravitational effects, e.g., lunar tides, that we wouldn't classify as gravitational waves.