Cooperative Positioning (CP) relies on a network of connected agents equipped with sensing and communication technologies to improve the positioning performance of standalone solutions. In this paper, we develop a completely data-driven model combining Long Short-Term Memory (LSTM) and Message Passing Neural Network (MPNN) for CP, where agents estimate their states from inter-agent and ego-agent measurements. The proposed LSTM-MPNN model is derived by exploiting the analogy with the probability-based Message Passing Algorithm (MPA), from which the graph-based structure of the problem and message passing scheme are inherited. In our solution, the LSTM block predicts the motion of the agents, while the MPNN elaborates the node and edge embeddings for an effective inference of the agents’ state. We present numerical evidence that our approach can enhance position estimation, while being at the same time an order of magnitude less complex than typical particle-based implementations of MPA for non-linear problems. In particular, the presented LSTM-MPNN model can reduce the error on agents’ positioning to one third compared to MPA-based CP, it holds a higher convergence speed and better exploits cooperation among agents.