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We study the ability of language models to reason about appropriate information disclosure - a central aspect of the evolving field of agentic privacy. Whereas previous works have focused on evaluating a model's ability to align with human decisions, we examine the role of ambiguity and missing context on model performance when making information-sharing decisions. We identify context ambiguity as a crucial barrier for high performance in privacy assessments. By designing Camber, a framework for context disambiguation, we show that model-generated decision rationales can reveal ambiguities and that systematically disambiguating context based on these rationales leads to significant accuracy improvements (up to 13.3% in precision and up to 22.3% in recall) as well as reductions in prompt sensitivity. Overall, our results indicate that approaches for context disambiguation are a promising way forward to enhance agentic privacy reasoning.