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As multi-agent systems proliferate, there is increasing need for coordination protocols that protect agents' sensitive information while still allowing them to collaborate. Often, a network system and controller are first designed and implemented, and then privacy is only incorporated after that. However, the absence of privacy from the design process can make it difficult to implement without significantly harming system performance. To help address this need, this paper presents a co-design framework for multi-agent networks and private controllers that we apply to the problem of private formation control. Agents' state trajectories are protected using differential privacy, which is a statistical notion of privacy that protects data by adding noise to it. Privacy noise alters the performance of the network, which we quantify by computing a bound for the steady-state error for private formations. Then, we analyze tradeoffs between privacy level, system performance, and connectedness of the network's communication topology. These trade-offs are used to formulate a co-design optimization framework to design the optimal communication topology alongside the optimal privacy parameters for a network running differentially private formation control. Simulation results illustrate the scalability of our proposed privacy/network co-design problem to large multi-agent networks, as well as the high quality of formations one can attain, even with privacy implemented.