Identifying informative components in binary data is an essential task in many research areas, including life sciences, social sciences, and recommendation systems. Boolean matrix factorization (BMF) is a family of methods that performs this task by efficiently factorizing the data. In real-world settings, the data is often distributed across stakeholders and required to stay private, prohibiting the straightforward application of BMF. To adapt BMF to this context, we approach the problem from a federated-learning perspective, while building on a state-of-the-art continuous binary matrix factorization relaxation to BMF that enables efficient gradient-based optimization. We propose to only share the relaxed component matrices, which are aggregated centrally using a proximal operator that regularizes for binary outcomes. We show the convergence of our federated proximal gradient descent algorithm and provide differential privacy guarantees. Our extensive empirical evaluation demonstrates that our algorithm outperforms, in terms of quality and efficacy, federation schemes of state-of-the-art BMF methods on a diverse set of real-world and synthetic data.