In reality you almost always find that the particles prefer to go to the same state because of some tiny energy shifts in the system. For example, a ferromagnetic condensate can have many degenerate states, but there is an energy cost for particles that disagree. These systems will break symmetry by having all the particles choose the same (arbitrary) state.
One exception is if the number of degenerate states is large, for instance larger than the number of particles. I have heard that claim that if the system has a sufficiently large degeneracy, for instance if all momentum states has the same energy, Bose-Einstein condensation might not occur.
Other than that, if there really is no energy cost to arranging your particles however you want between two degenerate states, I would guess that the particles would on average be equally distributed. However, fluctuations should be enormous: the probability of any configuration (N atoms in state 1, M atoms in state 2) should all be equal.