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As multi-agent systems proliferate, there is increasing demand for coordination protocols that protect agents' sensitive information while allowing them to collaborate. To help address this need, this paper presents a differentially private formation control framework. Agents' state trajectories are protected using differential privacy, which is a statistical notion of privacy that protects data by adding noise to it. We provide a private formation control implementation and analyze the impact of privacy upon the system. Specifically, we quantify tradeoffs between privacy level, system performance, and connectedness of the network's communication topology. These tradeoffs are used to develop guidelines for calibrating privacy in terms of control theoretic quantities, such as steady-state error, without requiring in-depth knowledge of differential privacy. Additional guidelines are also developed for treating privacy levels and network topologies as design parameters to tune the network's performance. Simulation results illustrate these tradeoffs and show that strict privacy is inherently compatible with strong system performance.