The problem of decentralized multi-robot target tracking asks for jointly selecting actions, e.g., motion primitives, for the robots to maximize target tracking performance with local communications. One major challenge for practical implementations is to make target tracking approaches scalable for large-scale problem instances. In this work, we propose a general-purpose learning architecture toward collaborative target tracking at scale, with decentralized communications. Particularly, our learning architecture leverages a graph neural network (GNN) to capture local interactions of the robots and learns decentralized decision-making for the robots. We train the learning model by imitating an expert solution and implement the resulting model for decentralized action selection involving local observations and communications only. We demonstrate the performance of our GNN-based learning approach in a scenario of active target tracking with large networks of robots. The simulation results show our approach nearly matches the tracking performance of the expert algorithm, and yet runs several orders faster with up to 100 robots. Moreover, it slightly outperforms a decentralized greedy algorithm but runs faster (especially with more than 20 robots). The results also exhibit our approach's generalization capability in previously unseen scenarios, e.g., larger environments and larger networks of robots.