  Motivated by the hypothesis that dark matter self-interactions provide a
solution to the small-scale structure formation problems, we investigate the
possibilities that the relic density of a self-interacting dark matter
candidate can proceed from the thermal freeze-out of annihilations into
Standard Model particles. We find that scalar and Majorana dark matter in the
mass range of $10-500$ MeV, coupled to a slightly heavier massive gauge boson,
are the only possible candidates in agreement with multiple current
experimental constraints. Here dark matter annihilations take place at a much
slower rate than the self-interactions simply because the interaction
connecting the Standard Model and the dark matter sectors is small. We also
discuss prospects of establishing or excluding these two scenarios in future
experiments.
