Centralized Training with Decentralized Execution (CTDE) has recently emerged as a popular framework for cooperative Multi-Agent Reinforcement Learning (MARL), where agents can use additional global state information to guide training in a centralized way and make their own decisions only based on decentralized local policies. Despite the encouraging results achieved, CTDE makes an independence assumption on agent policies, which limits agents from adopting global cooperative information from each other during centralized training. Therefore, we argue that the existing CTDE framework cannot fully utilize global information for training, leading to an inefficient joint exploration and perception, which can degrade the final performance. In this paper, we introduce a novel Centralized Advising and Decentralized Pruning (CADP) framework for MARL, that not only enables an efficacious message exchange among agents during training but also guarantees the independent policies for decentralized execution. Firstly, CADP endows agents the explicit communication channel to seek and take advice from different agents for more centralized training. To further ensure the decentralized execution, we propose a smooth model pruning mechanism to progressively constrain the agent communication into a closed one without degradation in agent cooperation capability. Empirical evaluations on different benchmarks and across various MARL backbones demonstrate that the proposed framework achieves superior performance compared with the state-of-the-art counterparts. Our code is available at https://github.com/zyh1999/CADP