In this brief paper, we investigate online training of Long Short Term Memory (LSTM) architectures in a distributed network of nodes, where each node employs an LSTM based structure for online regression. In particular, each node sequentially receives a variable length data sequence with its label and can only exchange information with its neighbors to train the LSTM architecture. We first provide a generic LSTM based regression structure for each node. In order to train this structure, we put the LSTM equations in a nonlinear state space form for each node and then introduce a highly effective and efficient Distributed Particle Filtering (DPF) based training algorithm. We also introduce a Distributed Extended Kalman Filtering (DEKF) based training algorithm for comparison. Here, our DPF based training algorithm guarantees convergence to the performance of the optimal LSTM coefficients in the mean square error (MSE) sense under certain conditions. We achieve this performance with communication and computational complexity in the order of the first order gradient based methods. Through both simulated and real life examples, we illustrate significant performance improvements with respect to the state of the art methods.