LLM agents increasingly act on behalf of users by selecting tools and constructing API requests to external services. This creates a new privacy risk in agentic systems: disclosure is no longer limited to what users directly enter into a form, but can instead be generated by the agent at runtime. In conventional web settings, disclosure is largely bounded by the user-facing interface, and what is appropriate to share varies across service contexts. In tool-using agents, however, disclosure is generated at runtime when user intent is translated into tool-call arguments for a particular receiving service, making context-sensitive disclosure boundaries harder to preserve. In this position paper, we argue that the runtime tool call is the key unit of privacy analysis in agentic systems. Our contribution is diagnostic rather than behavioral: instead of measuring realized leakage, we analyze interface conditions that may make agent oversharing more plausible. In particular, schemas that expose generic, weakly constrained free-text fields leave part of disclosure under agent discretion. In a case study of 2,344 tool specifications from the OpenAI GPT ecosystem, we find that 36.9% expose at least one such channel, creating conditions for within-context over-disclosure, cross-context leakage, and what we call contextual flattening. We conclude by outlining a research agenda for NLP that moves beyond output-only evaluation toward argument-level analysis of what tool schemas allow agents to send to third-party services.