As GenAI platforms grow, their dependence on content from competing providers, combined with access to alternative data sources, creates new challenges for data-sharing decisions. In this paper, we provide a model of data sharing between a content creation firm and a GenAI platform that can also acquire content from third-party experts. The interaction is modeled as a Stackelberg game: the firm first decides how much of its proprietary dataset to share with GenAI, and GenAI subsequently determines how much additional data to acquire from external experts. Their utilities depend on user traffic, monetary transfers, and the cost of acquiring additional data from external experts. We characterize the unique subgame perfect equilibrium of the game and uncover a surprising phenomenon: The firm may be willing to pay GenAI to share the firm's own data, leading to a costly data-sharing equilibrium. We further characterize the set of Pareto improving data prices, and show that such improvements occur only when the firm pays to share data. Finally, we study how the price can be set to optimize different design objectives, such as promoting firm data sharing, expert data acquisition, or a balance of both. Our results shed light on the economic forces shaping data-sharing partnerships in the age of GenAI, and provide guidance for platforms, regulators and policymakers seeking to design effective data exchange mechanisms.