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 H i , and compare our results with observations . The amount of material in the hot halo ( M _ { hot } \approx 4 - 5 \times 10 ^ { 10 } \mbox { M$ { } _ { \odot } $ } ) , and the cold ( M _ { cold } ( r \lesssim 10 { kpc } ) \approx 10 ^ { 8 } \mbox { M$ { } _ { \odot } $ } ) and H i ( M _ { HI } ( r \lesssim 50 { kpc } ) \approx 3 - 4 \times 10 ^ { 8 } \mbox { M$ { } _ { \odot% } $ } ) components display a reasonable agreement with observations . We also compute the accretion/ejection rates together with the H i ( radial and all-sky ) covering fractions . The integrated H i accretion rate within r = 50 kpc gives \sim 0.2 - 0.3 \mbox { 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 \lesssim 10 ^ { 5 } K can contribute significantly too . The net accretion rates of all material at the virial radii are 6 - 8 \mbox { 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 \sim 1 , as a consequence of the kinematical evolution of the LG .