Communication constraints often complicate group decision-making. In this experiment, we investigate how the network position of opinionated group members determines both the speed and the outcome of group consensus in 7-member communication networks susceptible to polarization. To this end, we implemented an online version of a color coordination task within experimentally controlled communication networks. In 72 networks, one individual was incentivized to prefer one of two options. In 156 networks, two individuals were incentivized to prefer conflicting options. The network positions of incentivized individuals were varied. In networks with a single incentivized individual, network position played no significant role in either the speed or outcome of consensus decisions. For conflicts, the incentivized individual with more neighbors was more likely to sway the group to their preferred outcome. Furthermore, consensus emerged more slowly when the opponents had the same number of neighbors, but could not see each other’s votes directly. These results suggest that the visibility of an opinion is key to wielding group influence, and that specific structures are sufficient to run communication networks into polarization, hindering a speedy consensus.