Running large language models (LLMs) locally is often considered more private than cloud-hosted inference because user prompts remain on the device. We ask whether keeping inference local is, by itself, sufficient to keep those prompts confidential. Our results show that it is not: prompt confidentiality also depends on how the surrounding serving software handles prompt data before, during, and after inference. We examine four boundaries at which prompt confidentiality can fail in consumer local-LLM serving systems: model loading, runtime memory, wrapper-level persistence, and the serving interface. To study these boundaries, we develop LLAnalyzer, a measurement framework that tests each boundary separately and traces observed failures to the responsible software component. Applying LLAnalyzer to four open-weight model families and two consumer deployment platforms, we find markedly different behaviour across boundaries. In a 24-hour AFL++ campaign with more than 12 million executions, we observe no parser crashes or successful malformed GGUF loads within the explored state space. Runtime memory tells a different story: we recover prompts after inference because multiple plaintext representations survive in allocator-managed memory, and sanitisation reduces this residue without eliminating it. We also find that consumer wrappers can extend prompt lifetime through plaintext persistence. At the serving boundary, we uncover a previously undocumented authorization flaw in llama.cpp that allows one authenticated client to restore another tenant's saved conversation state; the attack succeeds in 200/200 controlled trials. Separately, shared prompt-prefix caching exposes a remote timing oracle that remains distinguishable under WAN conditions. We argue that local LLM systems need explicit guarantees for prompt lifetime, persistent storage, and tenant isolation.

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