  Self-interacting dark matter (SIDM) is a simple and well-motivated scenario
that could explain long-standing puzzles in structure formation on small
scales. If the required self-interaction arises through a light mediator (with
mass $\sim 10$ MeV) in the dark sector, this new particle must be unstable to
avoid overclosing the universe. The decay of the light mediator could happen
due to a weak coupling of the hidden and visible sectors, providing new
signatures for direct detection experiments. The SIDM nuclear recoil spectrum
is more peaked towards low energies compared to the usual case of contact
interactions, because the mediator mass is comparable to the momentum transfer
of nuclear recoils. We show that the SIDM signal could be distinguished from
that of DM particles with contact interactions by considering the time-average
energy spectrum in experiments employing different target materials, or the
average and modulated spectra in a single experiment. Using current limits from
LUX and SuperCDMS, we also derive strong bounds on the mixing parameter between
hidden and visible sector.
