A major challenge as we move towards building agents for real-world problems, which could involve a massive number of human and/or machine agents, is that we must learn to reason about the behavior of these many other agents. In this paper, we consider the problem of scaling a predictive Theory of Mind (ToM) model to a very large number of interacting agents with a fixed computational budget. Motivated by the limited diversity of agent types, existing approaches to scalable TOM learn versatile single-agent representations for quickly adapting to new agents encountered sequentially. We consider the more general setting that many agents are observed in parallel and formulate the corresponding Theory of Many Minds (ToMM) problem of estimating the joint policy. We frame the scaling behavior of solutions in terms of parameter sharing schemes and in particular propose two parameter-free architectural features that endow models with the ability to exploit action correlations: encoding a multi-agent context, and decoding through an abstracted joint action space. The increased predictive capabilities that have come with foundation models have made it easier to imagine the possibility of using these models to make simulations that imitate the behavior of many agents within complex real-world systems. Being able to perform these simulations in a general-purpose way would not only help make more capable agents, it also would be a very useful capability for applications in social science, political science, and economics.