Professional networks -- the social networks among people in a given line of work -- can serve as a conduit for job prospects and other opportunities. Here we propose a model for the formation of such networks and the transfer of opportunities within them. In our theoretical model, individuals strategically connect with others to maximize the probability that they receive opportunities from them. We explore how professional networks balance connectivity, where connections facilitate opportunity transfers to those who did not get them from outside sources, and congestion, where some individuals receive too many opportunities from their connections and waste some of them. We show that strategic individuals are over-connected at equilibrium relative to a social optimum, leading to a price of anarchy for which we derive nearly tight asymptotic bounds. We also show that, at equilibrium, individuals form connections to those who provide similar benefit to them as they provide to others. Thus, our model provides a microfoundation in professional networking contexts for the fundamental sociological principle of homophily, that "similarity breeds connection," which in our setting is realized as a form of status homophily based on alignment in individual benefit. We further explore how, even if individuals are a priori equally likely to receive opportunities from outside sources, equilibria can be unequal, and we provide nearly tight bounds on how unequal they can be. Finally, we explore the ability for online platforms to intervene to improve social welfare and show that natural heuristics may result in adverse effects at equilibrium. Our simple model allows for a surprisingly rich analysis of coordination problems in professional networks and suggests many directions for further exploration.