Channel state information (CSI) in cellular networks is primarily used to maintain quality of service (QoS), yet it leaks privacy-sensitive information, in particular the user equipment's (UE) location. CSI is reported in plaintext on a publicly available schedule; thus, capturing it on the air and using a well trained AI model to localize the UE is straightforward. This simplicity in the attack execution makes it a serious privacy vulnerability. Using conventional privacy enhancing techniques, such as additive perturbation from differential privacy, to protect location data can potentially degrade the downlink communication quality. We present HideOut, a client-side defense mechanism that preserves the user's location from neural-based localization methods, while maintaining their downlink Spectral Efficiency (SE). Unlike generic adversarial attacks, HideOut treats downlink SE as a strict constraint in an optimization setting. Under maximum ratio transmission (MRT) beamforming, we show that a user-selected SE loss budget can be converted into an angular rate-feasible constraint on the reported channel vector. HideOut then performs gradient-based localization obfuscation while projecting each update onto this rate-feasible set. We evaluate HideOut against four CSI-based localization architectures under both white-box and black-box assumptions. Across models, HideOut increases localization ambiguity (error) from approximately $1--2$ m to $72--85$ m while keeping downlink SE loss within a $1\%$ budget. These results show that CSI-based localization can be substantially degraded at negligible communication quality loss when adversarial perturbations are made aware of the wireless link.

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