Autonomous LLM-based GUI agents demonstrate the potential to revolutionize mobile task automation, but they also introduce significant privacy risks due to their comprehensive access to on-screen user data. Standard architectures often leak Personally Identifiable Information (PII) by sending raw UI data to external LLM services. We address this vulnerability by designing, implementing, and evaluating \textit{PrivAuto}, a mobile GUI agent framework featuring a \textit{Perceive-Scan-Plan-Execute} loop. The core privacy-preserving ability lies in the on-device \textit{Scan} stage, which uses a lightweight model to sanitize UI data before it leaves the device, creating a practical privacy firewall. Empirical evaluation on the \textit{AndroidWorld} benchmark demonstrates the feasibility of this approach but reveals critical challenges. While effective at redacting Critical PII (>90% accuracy), the module struggles with Contextual PII (<30%) and is prone to failures in some typical scenarios. These findings offer crucial insights into the complex trade-offs between privacy and usability, highlighting the need for domain-specific PII identification models and robust engineering to ensure trustworthy LLM-based GUI agents.