Clusters of unvaccinated individuals are at risk of outbreaks of infection. When an individual's decision to choose vaccination is influenced by the choices of his social group, such clusters can readily arise. However, when the interactions that influence decision-making and those that permit the transmission of infection are different--for instance, when parents make vaccination decisions on behalf of their children--it is unclear how large the impact of this social influence will be. Here we use a modelling approach to represent social influence within a network of parents and the transmission of infection through a network of children. We show that the effect of social influence depends on the amount of overlap between the two different networks; large overlap means that clusters of parents who choose not to vaccinate are likely to have interacting children, generating clusters of unvaccinated children. Spatially local connections can further increase the impact of social influence. Outbreaks are most likely when parents who do not vaccinate have children who interact.