Consensus decision-making in social groups strongly depends on communication links that determine to whom individuals send, and from whom they receive, information. Here, we ask how consensus decisions are affected by strategic updating of links and how this effect varies with the direction of communication. We quantified the coevolution of link and opinion dynamics in a large population with binary opinions using mean-field numerical simulations of two voter-like models of opinion dynamics: an incoming model (IM) (where individuals choose who to receive opinions from) and an outgoing model (OM) (where individuals choose who to send opinions to). We show that individuals can bias group-level outcomes in their favour by breaking disagreeing links while receiving opinions (IM) and retaining disagreeing links while sending opinions (OM). Importantly, these biases can help the population avoid stalemates and achieve consensus. However, the role of disagreement avoidance is diluted in the presence of strong preferences—highly stubborn individuals can shape decisions to favour their preferences, giving rise to non-consensus outcomes. We conclude that collectively changing communication structures can bias consensus decisions, as a function of the strength of preferences and the direction of communication.