  We study the gas distribution in the Milky Way and Andromeda using a
constrained cosmological simulation of the Local Group (LG) within the context
of the CLUES (Constrained Local UniversE Simulations) project. We analyse the
properties of gas in the simulated galaxies at $z=0$ for three different
phases: `cold', `hot' and HI, and compare our results with observations. The
amount of material in the hot halo ($M_{hot}\approx
4-5\times10^{10}\,$M$_{\odot}$), and the cold
($M_{cold}(r\lesssim10\,$kpc$)\approx10^{8}\,$M$_{\odot}$) and HI
($M_{HI}(r\lesssim50\,$kpc$)\approx 3-4\times10^8\,$M$_{\odot}$) components
display a reasonable agreement with observations. We also compute the
accretion/ejection rates together with the HI (radial and all-sky) covering
fractions. The integrated HI accretion rate within $r=50\,$kpc gives
$\sim$$0.2-0.3\,$M$_{\odot}\,$yr$^{-1}$, i.e. close to that obtained from
high-velocity clouds in the Milky Way. We find that the global accretion rate
is dominated by hot material, although ionized gas with $T\lesssim10^5\,$K can
contribute significantly too. The $net$ accretion rates of $all$ material at
the virial radii are $6-8\,$M$_{\odot}\,$yr$^{-1}$. At $z=0$, we find a
significant gas excess between the two galaxies, as compared to any other
direction, resulting from the overlap of their gaseous haloes. In our
simulation, the gas excess first occurs at $z\sim1$, as a consequence of the
kinematical evolution of the LG.
