We study competitive data markets in which consumers own their personal data and can trade it with intermediaries, such as e-commerce platforms. Intermediaries use this data to provide services to the consumers, such as targeted offers from third-party merchants. Our main results identify a novel inefficiency, resulting in equilibrium data allocations that fail to maximize welfare. This inefficiency hinges on the role that intermediaries play as information gatekeepers, a hallmark of the digital economy. We provide three solutions to this market failure: establishing data unions, which manage consumers' data on their behalf; taxing the trade of data; and letting the price of data depend on its intended use.