In current power markets, the bulk of electricity is sold wholesale and transported to consumers via long-distance transmission lines. Recently, decentralized local power markets have evolved, often as isolated networks based on solar generation. We analyze strategic pricing, investment, and welfare in local power markets. We show that local power markets with peer-to-peer trading are competitive and provide efficient investment incentives, even for a small number of participating households. We identify positive network externalities that make larger markets more attractive but lead to inefficiencies where networks compete. Collectively, our results present a set of positive efficiency results for peer-to-peer electricity markets.