How diverse are sharing economy platforms? Are they fair marketplaces, where all participants operate on a level playing field, or are they large-scale online aggregators of offline human biases? Often portrayed as easy-to-access digital spaces whose participants receive equal opportunities, such platforms have recently come under fire due to reports of discriminatory behaviours among their users, and have been associated with gentrification phenomena that exacerbate preexisting inequalities along racial lines. In this paper, we focus on the Airbnb sharing economy platform, and analyse the diversity of its user base across five large cities. We find it to be predominantly young, female, and white. Notably, we find this to be true even in cities with a diverse racial composition. We then introduce a method based on the statistical analysis of networks to quantify behaviours of homophily, heterophily and avoidance between Airbnb hosts and guests. Depending on cities and property types, we do find signals of such behaviours relating both to race and gender. We use these findings to provide platform design recommendations, aimed at exposing and possibly reducing the biases we detect, in support of a more inclusive growth of sharing economy platforms.