In this paper, we are interested in understanding the relation between gossip and two social phenomena: consensus and the positivity bias. These two properties of a population together seem somewhat counterintuitive: a population needs some consensus to act as a group, at the same time the positivity bias is said to be quite universal and it means people diverge. This paradox can perhaps be solved by the understanding of its links to gossip and its social functions (Foster, 2004). Deffuant, Carletti, and Huet (2013) have shown that the Leviathan model is able to exhibit these two social phenomena. They emerge from the individual’s need to form a valuation of themselves (i.e. self-valuation), as well as defining the value of others, through direct interaction and gossip. The particular role of gossip in their emergence and maintenance has not been exhaustively investigated in this model. That is the purpose of this paper, which starts from four hypotheses: gossip leads to consensus which increases with its intensity; gossip decreases the strength of the positivity bias and can suppress it; positivity bias and disagreement are linked to each other; the positivity bias and bias to negativity occurring in the Leviathan model appear conjointly whatever the level of gossip (they have been conjointly diagnosed in the first investigations of Deffuant et al. 2013). Overall, our hypotheses are confirmed. We especially find that consensus is almost never reached without gossip. We also show how an asymmetrical level of openness to the influence of others, depending on how high they are held in esteem, is important for the positivity bias, as well as for the bias to negativity. The number of peers discussed during a meeting is also shown to be essential for consensus.