We investigate the extent and the properties of the Mg II cool, low-density absorbing gas located in the halo and in the circumgalactic environment of quasars, using a sample of 31 projected quasar pairs with impact parameter pd < 200 kpc in the redshift range 0.5 ≲ z ≲ 1.6. In the transverse direction, we detect 18 Mg II absorbers associated with the foreground quasars, while no absorption system originated by the gas surrounding the quasar itself is found along the line of sight. This suggests that the quasar emission induces an anisotropy in the absorbing gas distribution. Our observations indicate that the covering fraction (f_C) of Mg II absorption systems with rest-frame equivalent width W_r(λ2796) > 0.3 Å ranges from f_C ˜ 1.0 at pd ≲ 65 kpc to f_C ˜ 0.2 at pd ≳ 150 kpc, and appears to be higher than that for galaxies. Our findings support a scenario where the luminosity/mass of the host galaxies affect the extent and the richness of the absorbing Mg II circumgalactic medium.