We develop a model of opinion dynamics where agents in a social network seek to learn a ground truth among a set of competing hypotheses. Agents in the network form private beliefs about such hypotheses by aggregating their neighbors' publicly stated beliefs, in an iterative fashion. This process allows us to keep track of scenarios where private and public beliefs align, leading to population-wide consensus on the ground truth, as well as scenarios where the two sets of beliefs fail to converge. The latter scenario - which is reminiscent of the phenomenon of cognitive dissonance - is induced by injecting 'conspirators' in the network, i.e., agents who actively spread disinformation by not communicating accurately their private beliefs. We show that the agents' cognitive dissonance non-trivially reaches its peak when conspirators are a relatively small minority of the population, and that such an effect can be mitigated - although not erased - by the presence of 'debunker' agents in the network.