Constructing proper representations of molecules lies at the core of numerous tasks such as molecular property prediction and drug design. Graph neural networks, especially message passing neural network (MPNN) and its variants, have recently made remarkable achievements in molecular graph modeling. Albeit powerful, the one-sided focuses on atom (node) or bond (edge) information of existing MPNN methods lead to the insufficient representations of the attributed molecular graphs. Herein, we propose a Communicative Message Passing Neural Network (CMPNN) to improve the molecular embedding by strengthening the message interactions between nodes and edges through a communicative kernel. In addition, the message generation process is enriched by introducing a new message booster module. Extensive experiments demonstrated that the proposed model obtained superior performances against state-of-the-art baselines on six chemical property datasets. Further visualization also showed better representation capacity of our model.