Existing works on identification of the dynamics of linear time-invariant (LTI) systems primarily focus on the least squares (LS) method when the recorded trajectories are rich and satisfy conditions such as the persistency of excitation. In this paper, we consider the case where the recorded states and inputs are not sufficiently rich, and present a system identification framework -- inspired by multi-task learning -- that estimates the matrices of a given number of LTI systems jointly, by leveraging structural similarities across the LTI systems. By regularizing the LS fit for each system with a function that enforces common structural properties, the proposed method alleviates the ill-conditioning of the LS when the recorded trajectories are not sufficiently rich. We consider priors where, for example, the LTI systems are similar in the sense that the system matrices share a common sparsity pattern, some matrices are linear combinations of others, or their norm difference is small. We outline a proximal-gradient method to solve the multi-task identification problem, and we propose a decentralized algorithm in the spirit of existing federated learning architectures. We provide empirical evidence of the effectiveness of the proposed method by considering a synthetic dataset, and by applying our method to the problem of estimating the dynamics of brain networks. For the latter, the proposed method requires a significantly smaller number of fMRI readings to achieve similar error levels of the LS when estimating the brain dynamics across subjects.