The literature on social networks often presumes that job search through (strong) social ties leads to increased inequality by providing privileged individuals with access to more attractive labor market opportunities. We assess this presumption in the context of sorting between AKM-style person and establishment fixed effects. Our rich Swedish register data allow us to measure connections between agents - workers to workers and workers to firms - through parents, children, siblings, spouses, former co-workers and classmates from high school/college, and current neighbors. In clear contrast with the above presumption, there is less sorting inequality among the workers hired through social networks. This outcome results from opposing factors. On the one hand, reinforcing positive sorting, high-wage job seekers are shown to have social connections to high-wage workers, and therefore to high-wage firms (because of sorting of workers over firms). Furthermore, connections have a causal impact on the allocation of workers across workplaces - employers are much more likely to hire displaced workers to whom they are connected through their employees, in particular if their social ties are strong. On the other hand, attenuating positive sorting, the (causal) impact is much stronger for low-wage firms than it is for high-wage firms, irrespective of the type of worker involved, even conditional on worker fixed effects. The lower degree of sorting among connected hires thus arises because low-wage firms use their (relatively few) connections to high-wage workers to hire workers of a type that they are unable to attract through market channels.