I consider a model of markets for personal data, where data intermediaries (e.g., online platforms and data brokers) buy data from consumers and sell them to downstream firms. Competition among intermediaries has a limited impact on improving consumer welfare: If intermediaries offer high prices for data, consumers share data with multiple intermediaries, which lowers the downstream price of data and hurts intermediaries. This leads to multiple equilibria. There is a monopoly equilibrium, and an equilibrium with greater data concentration benefits intermediaries and hurts consumers. I generalize the results to arbitrary consumer preferences and study information design by data intermediaries.