This paper proposes a cooperative integrated sensing and communication network (Co-ISACNet) adopting hybrid beamforming (HBF) architecture, which improves both radar sensing and communication performance. The main contributions of this work are four-fold. First, we introduce a novel cooperative sensing method for the considered Co-ISACNet, followed by a comprehensive analysis of this method. This analysis mathematically verifies the benefits of Co-ISACNet and provides insightful design guidelines. Second, to show the benefits of Co-ISACNet, we propose to jointly design the HBF to maximize the network communication capacity while satisfying the constraint of beampattern similarity for radar sensing, which results in a highly dimensional and non-convex problem. Third, to facilitate the joint design, we propose a novel distributed optimization framework based on proximal gradient and alternating direction method of multipliers, namely PANDA. Fourth, we further adopt the proposed PANDA framework to solve the joint HBF design problem for the Co-ISACNet. By using the proposed PANDA framework, all access points (APs) optimize the HBF in parallel, where each AP only requires local channel state information and limited message exchange among the APs. Such framework reduces significantly the computational complexity and thus has pronounced benefits in practical scenarios. Simulation results verify the effectiveness of the proposed algorithm compared with the conventional centralized algorithm and show the remarkable performance improvement of radar sensing and communication by deploying Co-ISACNet.