Multi-agent Large Language Model (LLM) systems create privacy risks that current benchmarks cannot measure. When agents coordinate on tasks, sensitive data passes through inter-agent messages, shared memory, and tool arguments, all pathways that output-only audits never inspect. We introduce AgentLeak, to the best of our knowledge the first full-stack benchmark for privacy leakage covering internal channels. It spans 1,000 scenarios across healthcare, finance, legal, and corporate domains, paired with a 32-class attack taxonomy and a three-tier detection pipeline. A factorial evaluation crossing five production LLMs (GPT-4o, GPT-4o-mini, Claude 3.5 Sonnet, Mistral Large, and Llama 3.3 70B) with all 1,000 scenarios, yielding 4,979 validated execution traces, reveals that multi-agent configurations reduce per-channel output leakage (C1: 27.2\% vs 43.2\% in single-agent) but introduce unmonitored internal channels that raise total system exposure to 68.9\% (aggregated across C1, C2, C5). Internal channels account for most of this gap: inter-agent messages (C2) leak at 68.8\%, compared to 27.2\% on C1 (output channel). This means that output-only audits miss 41.7\% of violations. Safety-aligned models achieve lower leakage on both external and internal channels, yet no model eliminates it. Across all five models and four domains, the pattern C2 $\geq$ C1 holds consistently, confirming that inter-agent communication is the primary vulnerability. These results establish that output-only auditing is fundamentally insufficient for multi-agent systems and that privacy controls must be extended to inter-agent communication channels.