  We investigate the population of bright satellites ($M_{*} \geq 10^{5}
\mathrm{M}_{\odot}$) of haloes of mass comparable to that of the Milky Way in
cosmological simulations in which the dark matter (DM) is either cold, warm or
self-interacting (CDM, WDM and SIDM respectively). The nature of the DM gives
rise to differences in the abundance and structural properties of field halos.
In WDM, the main feature is a reduction in the total number of galaxies that
form, reflecting a suppression of low-mass DM haloes and lower galaxy formation
efficiency compared to CDM. For SIDM, the changes are structural, restricted to
the central regions of haloes and dependent on the assumed self-interaction
cross-section. We also consider different baryonic subgrid physics models for
galaxy formation, in which supernova gas blowouts can or cannot induce the
formation of a core in dwarf galaxies. Overall, the inclusion of baryons lessen
the differences in the halo properties in the different DM models compared to
DM-only simulations. This affects the satellite properties at infall and
therefore their subsequent tidal stripping and survival rates. Nonetheless, we
find slightly less concentrated satellite radial distributions as the SIDM
cross-section increases. Unfortunately, we also find that the satellite
populations in simulations with baryon-induced cores in CDM and WDM can mimic
the results found in SIDM, making the satellite stellar mass and maximum
circular velocity functions heavily degenerate on the assumed nature of the DM
and the adopted subgrid modelling. These degeneracies preclude using the
brightest satellites of the Milky Way to constrain the nature of DM.
