  (Abridged). The optical light curve of some SNe may be powered by the outward
diffusion of the energy deposited by the explosion shock in optically thick
circumstellar matter (CSM). Recently, it was shown that the radiation-mediated
and -dominated shock in an optically thick wind must transform into a
collisionless shock and can produce hard X-rays. The X-rays are expected to
peak at late times, relative to maximum visible light. Here we report on a
search, using Swift and Chandra, for X-ray emission from 28 SNe that belong to
classes whose progenitors are suspected to be embedded in dense CSM
(IIn/Ibn/SLSN-I). Two SNe in our sample have X-ray properties that are roughly
consistent with the expectation for X-rays from a collisionless shock in
optically thick CSM. Therefore, we suggest that their optical light curves are
powered by shock breakout in CSM. We show that two other events were too X-ray
bright during the SN maximum optical light to be explained by the shock
breakout model. We conclude that the light curves of some, but not all,
type-IIn/Ibn SNe are powered by shock breakout in CSM. For the rest of the SNe
in our sample, including all the SLSN-I events, our X-ray limits are not deep
enough and were typically obtained at too early times to conclude about their
nature. We argue that the optical light curves of SNe, for which the X-ray
emission peaks at late times, are likely powered by the diffusion of shock
energy from a dense CSM. We comment about the possibility to detect some of
these events in radio.
