A Large Language Model (LLM) powered GUI agent is a specialized autonomous system that performs tasks on the user's behalf according to high-level instructions. It does so by perceiving and interpreting the graphical user interfaces (GUIs) of relevant apps, often visually, inferring necessary sequences of actions, and then interacting with GUIs by executing the actions such as clicking, typing, and tapping. To complete real-world tasks, such as filling forms or booking services, GUI agents often need to process and act on sensitive user data. However, this autonomy introduces new privacy and security risks. Adversaries can inject malicious content into the GUIs that alters agent behaviors or induces unintended disclosures of private information. These attacks often exploit the discrepancy between visual saliency for agents and human users, or the agent's limited ability to detect violations of contextual integrity in task automation. In this paper, we characterized six types of such attacks, and conducted an experimental study to test these attacks with six state-of-the-art GUI agents, 234 adversarial webpages, and 39 human participants. Our findings suggest that GUI agents are highly vulnerable, particularly to contextually embedded threats. Moreover, human users are also susceptible to many of these attacks, indicating that simple human oversight may not reliably prevent failures. This misalignment highlights the need for privacy-aware agent design. We propose practical defense strategies to inform the development of safer and more reliable GUI agents.