This paper addresses two major challenges in terahertz (THz) channel estimation: the beam-split phenomenon, i.e., beam misalignment because of frequency-independent analog beamformers, and computational complexity because of the usage of ultra-massive number of antennas to compensate propagation losses. Data-driven techniques are known to mitigate the complexity of this problem but usually require the transmission of the datasets from the users to a central server entailing huge communication overhead. In this work, we introduce a federated multi-task learning (FMTL), wherein the users transmit only the model parameters instead of the whole dataset, for THz channel and user direction-of-arrival (DoA) estimation to improve the communications-efficiency. We first propose a novel beamspace support alignment technique for channel estimation with beam-split correction. Then, the channel and DoA information are used as labels to train an FMTL model. By exploiting the sparsity of the THz channel, the proposed approach is implemented with fewer pilot signals than the traditional techniques. Compared to the previous works, our FMTL approach provides higher channel estimation accuracy as well as approximately 25 (32) times lower model (channel) training overhead, respectively.