  Hybrid optomechanical systems are emerging as a fruitful architecture for
quantum technologies. Hence, determining the relevant atom-light and
light-mechanics couplings is an essential task in such systems. The fingerprint
of these couplings is left in the global state of the system during
non-equilibrium dynamics. However, in practice, performing measurements on the
entire system is not feasible, and thus, one has to rely on partial access to
one of the subsystems, namely the atom, the light, or the mechanics. Here, we
perform a comprehensive analysis to determine the optimal subsystem for probing
the couplings. We find that if the light-mechanics coupling is known or
irrelevant, depending on the range of the qubit-light coupling, then the
optimal subsystem can be either light or the qubit. In other scenarios, e.g.,
simultaneous estimation of the couplings, the light is usually the optimal
subsystem. This can be explained as light is the mediator between the other two
subsystems. Finally, we show that the widely used homodyne detection can
extract a fair fraction of the information about the couplings from the light
degrees of freedom.
