Adaptive Traffic Signal Control (ATSC) has become a popular research topic in intelligent transportation systems. Regional Traffic Signal Control (RTSC) using the Multi-agent Deep Reinforcement Learning (MADRL) technique has become a promising approach for ATSC due to its ability to achieve the optimum trade-off between scalability and optimality. Most existing RTSC approaches partition a traffic network into several disjoint regions, followed by applying centralized reinforcement learning techniques to each region. However, the pursuit of cooperation among RTSC agents still remains an open issue and no communication strategy for RTSC agents has been investigated. In this paper, we propose communication strategies to capture the correlation of micro-traffic states among lanes and the correlation of macro-traffic states among intersections. We first justify the evolution equation of the RTSC process is Markovian via a system of store-and-forward queues. Next, based on the evolution equation, we propose two GAT-Aggregated (GA2) communication modules--GA2-Naive and GA2-Aug to extract both intra-region and inter-region correlations between macro and micro traffic states. While GA2-Naive only considers the movements at each intersection, GA2-Aug also considers the lane-changing behavior of vehicles. Two proposed communication modules are then aggregated into two existing novel RTSC frameworks--RegionLight and Regional-DRL. Experimental results demonstrate that both GA2-Naive and GA2-Aug effectively improve the performance of existing RTSC frameworks under both real and synthetic scenarios. Hyperparameter testing also reveals the robustness and potential of our communication modules in large-scale traffic networks.