LLM-powered agents are beginning to automate user's tasks across the open web, often with access to user resources such as emails and calendars. Unlike standard LLMs answering questions in a controlled ChatBot setting, web agents act "in the wild", interacting with third parties and leaving behind an action trace. Therefore, we ask the question: how do web agents handle user resources when accomplishing tasks on their behalf across live websites? In this paper, we formalize Natural Agentic Oversharing -- the unintentional disclosure of task-irrelevant user information through an agent trace of actions on the web. We introduce SPILLage, a framework that characterizes oversharing along two dimensions: channel (content vs. behavior) and directness (explicit vs. implicit). This taxonomy reveals a critical blind spot: while prior work focuses on text leakage, web agents also overshare behaviorally through clicks, scrolls, and navigation patterns that can be monitored. We benchmark 180 tasks on live e-commerce sites with ground-truth annotations separating task-relevant from task-irrelevant attributes. Across 1,080 runs spanning two agentic frameworks and three backbone LLMs, we demonstrate that oversharing is pervasive with behavioral oversharing dominates content oversharing by 5x. This effect persists -- and can even worsen -- under prompt-level mitigation. However, removing task-irrelevant information before execution improves task success by up to 17.9%, demonstrating that reducing oversharing improves task success. Our findings underscore that protecting privacy in web agents is a fundamental challenge, requiring a broader view of "output" that accounts for what agents do on the web, not just what they type. Our datasets and code are available at https://github.com/jrohsc/SPILLage.