The cooperative behavior of quantum impurities on 2D materials, such as graphene and bilayer graphene, is characterized by a non-trivial competition between screening (Kondo effect), and Ruderman-Kittel-Kasuya-Yosida (RKKY) magnetism. In addition, due to the small density of states at the Fermi level, impurities may not couple to the conduction electrons at all, behaving as free moments. Employing a recently developed {em{exact}} numerical method to study multi-impurity lattice systems, we obtain non-perturbative results that dramatically depart from expectations based on the conventional RKKY theory. At half-filling and for weak coupling, impurities remain in the local moment regime when they are on opposite sublattices, up to a critical value of the interactions when they start coupling anti-ferromagnetically with correlations that decay very slowly with inter-impurity distance. At finite doping, away from half-filling, ferromagnetism is completely absent and the physics is dominated by a competition between anti-ferromagnetism and Kondo effect. In bilayer graphene, impurities on opposite layers behave as free moments, unless the interaction is of the order of the hopping or larger.
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