Deep research agents increasingly combine private local documents with external tools like web retrieval, creating a privacy risk: an agent's external queries may leak sensitive information from its local context. This risk is amplified by the mosaic effect, where individual queries may appear harmless but become revealing in aggregate. We introduce MosaicLeaks, a benchmark of 1,001 multi-hop deep research tasks that chain private enterprise documents and a public web corpus, forcing agents to make external queries that depend on local information. We evaluate leakage with an adversary LLM that observes only the agent's external queries and attempts to infer private information at three levels: the agent's research intent, answers to specific private questions and verifiable claims about the enterprise documents. We find that models across families and sizes frequently leak at all three levels, that zero-shot privacy prompting reduces but does not eliminate leakage and that reinforcement learning for task performance alone worsens leakage. To address this, we propose Privacy-Aware Deep Research (PA-DR), an RL framework that combines situational rewards for task success with a learned privacy classifier to provide dense credit assignment over both per-query and mosaic-level leakage. Training Qwen3-4B-Instruct with PA-DR improves accuracy from 48.7% to 58.7% and reduces answer and full-information leakage from 34.0% to 9.9%.