Recently, a series of algorithms have been explored for GAN compression, which aims to reduce tremendous computational overhead and memory usages when deploying GANs on resource-constrained edge devices. However, most of the existing GAN compression work only focuses on how to compress the generator, while fails to take the discriminator into account. In this work, we revisit the role of discriminator in GAN compression and design a novel generator-discriminator cooperative compression scheme for GAN compression, termed GCC. Within GCC, a selective activation discriminator automatically selects and activates convolutional channels according to a local capacity constraint and a global coordination constraint, which help maintain the Nash equilibrium with the lightweight generator during the adversarial training and avoid mode collapse. The original generator and discriminator are also optimized from scratch, to play as a teacher model to progressively refine the pruned generator and the selective activation discriminator. A novel online collaborative distillation scheme is designed to take full advantage of the intermediate feature of the teacher generator and discriminator to further boost the performance of the lightweight generator. Extensive experiments on various GAN-based generation tasks demonstrate the effectiveness and generalization of GCC. Among them, GCC contributes to reducing 80% computational costs while maintains comparable performance in image translation tasks.