In a Bernal graphene bilayer, carbon atoms belong to two inequivalent sublattices A and B, with atoms that are coupled to the other layer by [Formula: see text] bonds belonging to sublattice A and the other atoms belonging to sublattice B. We analyze the density of states and the conductivity of Bernal graphene bilayers when atoms of sublattice A or B only are randomly functionalized. We find that for a selective functionalization on sublattice B only, a mobility gap of the order of 0.5 eV is formed close to the Dirac energy at concentration of adatoms [Formula: see text]. In addition, at some other energies conductivity presents anomalous behaviors. We show that these properties are related to the bipartite structure of the graphene layer.