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 ($M_{hot}approx 4-5times10^{10},$M$_{odot}$), and the cold ($M_{cold}(rlesssim10,$kpc$)approx10^{8},$M$_{odot}$) and HI ($M_{HI}(rlesssim50,$kpc$)approx 3-4times10^8,$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 $Tlesssim10^5,$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 $zsim1$, as a consequence of the kinematical evolution of the LG.
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