Recent studies have used speech signals to assess depression. However, speech features can lead to serious privacy concerns. To address these concerns, prior work has used privacy-preserving speech features. However, using a subset of features can lead to information loss and, consequently, non-optimal model performance. Furthermore, prior work relies on a centralized approach to support continuous model updates, posing privacy risks. This paper proposes to use Federated Learning (FL) to enable decentralized, privacy-preserving speech analysis to assess depression. Using an existing dataset (DAIC-WOZ), we show that FL models enable a robust assessment of depression with only 4--6% accuracy loss compared to a centralized approach. These models also outperform prior work using the same dataset. Furthermore, the FL models have short inference latency and small memory footprints while being energy-efficient. These models, thus, can be deployed on mobile devices for real-time, continuous, and privacy-preserving depression assessment at scale.