Multi-exit networks improve inference efficiency by allowing inputs to terminate early at intermediate exits. We identify a privacy vulnerability in cloud deployments of these networks: a malicious Cloud Service Provider (CSP) can infer sensitive properties of individual user inputs solely from inference latency, without accessing inputs, intermediate states, or model outputs. The attack exploits the CSP's control over the model supply chain. Prior to deployment, the CSP supplies a specially crafted multi-exit network manipulated so that inputs carrying a target sensitive property are routed to a designated exit with high probability, while other inputs exit elsewhere. At serving time, the CSP infers the exit index from per-query latency, thereby determining whether each input possesses the target property without accessing inputs, outputs, or internal model states. We propose a training framework combining selective label smoothing and entropy-covariance optimization to control confidence distributions across exits and reduce premature termination. We conduct experiments on four datasets across ten target properties. The results demonstrate strong attack effectiveness and stealthiness. The attack achieves an average F1 score of 0.73, compared to 0.25 for the strongest baseline, while main task accuracy drops by less than 1.3\% and inference speedup degradation remains below 0.1 in most settings. Our code is available at https://github.com/kazefjj/Weaponizing_the_Exit.

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