Accurate positioning is known to be a fundamental requirement for the deployment of Connected Automated Vehicles (CAVs). To meet this need, a new emerging trend is represented by cooperative methods where vehicles fuse information coming from navigation and imaging sensors via Vehicle-to-Everything (V2X) communications for joint positioning and environmental perception. In line with this trend, this paper proposes a novel data-driven cooperative sensing framework, termed Cooperative LiDAR Sensing with Message Passing Neural Network (CLS-MPNN), where spatially-distributed vehicles collaborate in perceiving the environment via LiDAR sensors. Vehicles process their LiDAR point clouds using a Deep Neural Network (DNN), namely a 3D object detector, to identify and localize possible static objects present in the driving environment. Data are then aggregated by a centralized infrastructure that performs Data Association (DA) using a Message Passing Neural Network (MPNN) and runs the Implicit Cooperative Positioning (ICP) algorithm. The proposed approach is evaluated using two realistic driving scenarios generated by a high-fidelity automated driving simulator. The results show that CLS-MPNN outperforms a conventional non-cooperative localization algorithm based on Global Navigation Satellite System (GNSS) and a state-of-the-art cooperative Simultaneous Localization and Mapping (SLAM) method while approaching the performances of an oracle system with ideal sensing and perfect association.