We present a comparative study of oxygen intercalation under polycrystalline and highly ordered graphene samples on the Co(0001) surface, which are relevant for magnetic applications. A comprehensive experimental study by means of microscopic and spectroscopic techniques shows that intercalation leads to hole-doping and decoupling of graphene from the strongly bound cobalt surface. However, only for the polycrystalline graphene samples, the full intercalation was achieved. The kinetics of intercalation, graphene etching, and cobalt oxidation has been thoroughly investigated. The obtained results have allowed us to disclose the key points of the oxygen intercalation process for both highly ordered and strongly disordered graphene on cobalt.