Generating gestures from human speech has gained tremendous progress in animating virtual avatars. While the existing methods enable synthesizing gestures cooperated by individual self-talking, they overlook the practicality of concurrent gesture modeling with two-person interactive conversations. Moreover, the lack of high-quality datasets with concurrent co-speech gestures also limits handling this issue. To fulfill this goal, we first construct a large-scale concurrent co-speech gesture dataset that contains more than 7M frames for diverse two-person interactive posture sequences, dubbed GES-Inter. Additionally, we propose Co$^3$Gesture, a novel framework that enables coherent concurrent co-speech gesture synthesis including two-person interactive movements. Considering the asymmetric body dynamics of two speakers, our framework is built upon two cooperative generation branches conditioned on separated speaker audio. Specifically, to enhance the coordination of human postures with respect to corresponding speaker audios while interacting with the conversational partner, we present a Temporal Interaction Module (TIM). TIM can effectively model the temporal association representation between two speakers' gesture sequences as interaction guidance and fuse it into the concurrent gesture generation. Then, we devise a mutual attention mechanism to further holistically boost learning dependencies of interacted concurrent motions, thereby enabling us to generate vivid and coherent gestures. Extensive experiments demonstrate that our method outperforms the state-of-the-art models on our newly collected GES-Inter dataset. The dataset and source code are publicly available at \href{https://mattie-e.github.io/Co3/}{\textit{https://mattie-e.github.io/Co3/}}.