The proliferation of LLM-based agents has led to increasing deployment of inter-agent collaboration for tasks like scheduling, negotiation, resource allocation etc. In such systems, privacy is critical, as agents often access proprietary tools and domain-specific databases requiring strict confidentiality. This paper examines whether LLM-based agents demonstrate an understanding of contextual privacy. And, if instructed, do these systems preserve inference time user privacy in non-adversarial multi-turn conversation. Existing benchmarks to evaluate contextual privacy in LLM-agents primarily assess single-turn, low-complexity tasks where private information can be easily excluded. We first present a benchmark - MAGPIE comprising 158 real-life high-stakes scenarios across 15 domains. These scenarios are designed such that complete exclusion of private data impedes task completion yet unrestricted information sharing could lead to substantial losses. We then evaluate the current state-of-the-art LLMs on (a) their understanding of contextually private data and (b) their ability to collaborate without violating user privacy. Empirical experiments demonstrate that current models, including GPT-4o and Claude-2.7-Sonnet, lack robust understanding of contextual privacy, misclassifying private data as shareable 25.2\% and 43.6\% of the time. In multi-turn conversations, these models disclose private information in 59.9\% and 50.5\% of cases even under explicit privacy instructions. Furthermore, multi-agent systems fail to complete tasks in 71\% of scenarios. These results underscore that current models are not aligned towards both contextual privacy preservation and collaborative task-solving.