Despite the generative model's groundbreaking success, the need to study its implications for privacy and utility becomes more urgent. Although many studies have demonstrated the privacy threats brought by GANs, no existing survey has systematically categorized the privacy and utility perspectives of GANs and VAEs. In this article, we comprehensively study privacy-preserving generative models, articulating the novel taxonomies for both privacy and utility metrics by analyzing 100 research publications. Finally, we discuss the current challenges and future research directions that help new researchers gain insight into the underlying concepts.