Millimeter wave multiple-input multiple-output (mmWave-MIMO) systems with small number of radio-frequency (RF) chains have limited multiplexing gain. Spatial path index modulation (SPIM) is helpful in improving this gain by utilizing additional signal bits modulated by the indices of spatial paths. In this paper, we introduce model-based and model-free frameworks for beamformer design in multi-user SPIM-MIMO systems. We first design the beamformers via model-based manifold optimization algorithm. Then, we leverage federated learning (FL) with dropout learning (DL) to train a learning model on the local dataset of users, who estimate the beamformers by feeding the model with their channel data. The DL randomly selects different set of model parameters during training, thereby further reducing the transmission overhead compared to conventional FL. Numerical experiments show that the proposed framework exhibits higher spectral efficiency than the state-of-the-art SPIM-MIMO methods and mmWave-MIMO, which relies on the strongest propagation path. Furthermore, the proposed FL approach provides at least 10 times lower transmission overhead than the centralized learning techniques.