This paper studies the growth of online peer-to-peer markets. Using data from TaskRabbit, a rapidly expanding marketplace for domestic tasks, we show that growth is highly heterogeneous across cities, which is a common feature of peer-to-peer markets for local services. To disentangle the potential drivers of growth, we present a general model of a frictional market for services, and estimate it using variation in the number of buyers and sellers present on the marketplace over time and across cities. First, we find that supply is highly elastic: in periods when demand doubles, sellers work almost twice as hard, prices hardly increase and the probability of requested tasks being matched only slightly falls. The first result implies that in markets where supply can accommodate wide fluctuations in demand, growth relies on attracting buyers at a faster rate than sellers. Second, we show that cities where the market fundamentals promote efficient matching of buyers and sellers are also the cities that grow fast in the number of buyers. This heterogeneity in matching efficiency is not attributable to scale economies, but is instead related to two measures of market thickness: geographic density (buyers and sellers living close together), and level of task standardization (buyers requesting homogeneous tasks). The second result implies that when network effects are limited by the local and time-sensitive nature of the services exchanged, growth of peer-to-peer markets largely depends on strategic geographic expansion.