  The Andromeda (M31) and Triangulum (M33) galaxies are the closest Local Group
galaxies to the Milky Way, being only 785 and 870 kpc away. These two galaxies
provide an independent view of high-energy processes that are often obscured in
our own Galaxy, including possible signals of dark matter (DM) particle
interactions. The Fermi Large Area Telescope (Fermi-LAT) preliminary eight year
list of sources includes both M31, which is detected as extended with a size of
about 0.4$^\circ$, and M33, which is detected as a point-like source. The
spatial morphology of M31 $\gamma$-ray emission could trace a population of
unresolved sources and energetic particles originating in sources not related
to massive star formation. Alternatively, the $\gamma$-ray emission could also
be an indication of annihilation or decay of DM particles. We investigate these
two possibilities using almost 10 years of data from the Fermi LAT. An
interpretation that involves only a DM $\gamma$-ray emission is in tension with
the current limits from other searches, such as those targeting Milky Way dwarf
spheroidal galaxies. When we include a template of astrophysical emission,
tuned on $\gamma$-ray data or from observations of these galaxies in other
wavelengths, we do not find any significant evidence for a DM contribution and
we set limits for the annihilation cross section that probe the thermal cross
section for DM masses up to a few tens of GeV in the $b\bar{b}$ and
$\tau^+\tau^-$ channels. For models where the DM substructures have masses
above $10^{-6}$ solar masses our limits probe the DM interpretation of the
Fermi LAT Galactic center excess. We provide also the lower limit for the DM
decay time assuming the same spatial models of the DM distribution in M31 and
M33.
