A crucial building block of responsible artificial intelligence is responsible data governance, including data collection. Its importance is also underlined in the latest EU regulations. The data should be of high quality, foremost correct and representative, and individuals providing the data should have autonomy over what data is collected. In this paper, we consider the setting of collecting personally measured fitness data (physical activity measurements), in which some individuals may not have an incentive to measure and report accurate data. This can significantly degrade the quality of the collected data. On the other hand, high-quality collective data of this nature could be used for reliable scientific insights or to build trustworthy artificial intelligence applications. We conduct a framed field experiment (N = 691) to examine the effect of offering fixed and quality-dependent monetary incentives, on the quality of the collected data. We use a peer-based incentive-compatible mechanism for the quality-dependent incentives without spot-checking or surveilling individuals. We find that the incentive-compatible mechanism can elicit good quality data while providing a good user experience and compensating fairly, although, in the specific study context, the data quality does not necessarily differ under the two incentive schemes. We contribute new design insights from the experiment and discuss directions that future field experiments and applications on explainable and transparent data collection may focus on.