Although eye tracking brings many benefits to users of mobile devices and developers of mobile applications, it poses significant privacy risks to both: the users of mobile devices, and the bystanders that surround users, are within the front-facing camera's field of view. Recent research demonstrates that tracking an individual's gaze reveals personal and sensitive information. This paper presents an investigation of the privacy perceptions and the subjective acceptance of users towards eye tracking on handheld mobile devices. In a four-phase user study (N=17), participants used a smartphone eye tracking app, were interviewed before and after viewing a video showing the amount of sensitive and personal data that could be derived from eye movements, and had their privacy concerns measured. Our findings 1) show factors that influence users' and bystanders' attitudes toward eye tracking on mobile devices such as the algorithms' transparency and the developers' credibility and 2) support designing mechanisms to allow for privacy-aware eye tracking solutions on mobile-devices.