Critical thinking and skepticism are fundamental mechanisms that one may use to prevent the spreading of rumors, fake news, and misinformation. We consider a simple model in which agents without previous contact with the rumor, being skeptically oriented, may convince spreaders to stop their activity or, once exposed to the rumor, decide not to propagate it as a consequence, for example, of fact checking. We extend a previous, mean-field analysis of the combined effect of these two mechanisms, active and passive skepticism, to include spatial correlations. This can be done either analytically, through the pair approximation, or simulating an agent-based version on diverse networks. Our results show that while in mean field there is no coexistence between spreaders and susceptibles (although, depending on the parameters, there may be bistability depending on the initial conditions), when spatial correlations are included, because of the protective effect of the isolation provided by removed agents, coexistence is possible.