A new bimodal generative model is proposed for generating conditional and joint samples, accompanied with a training method with learning a succinct bottleneck representation.The proposed model, dubbed as the variational Wyner model, is designed based on two classical problems in network information theory---distributed simulation and channel synthesis---in which Wyner's common information arises as the fundamental limit on the succinctness of the common representation.The model is trained by minimizing the symmetric Kullback--Leibler divergence between variational and model distributions with regularization terms for common information, reconstruction consistency, and latent space matching terms, which is carried out via an adversarial density ratio estimation technique.