Personal data markets have become ubiquitous. At the same time, the non-rivalry of data suggests that the social returns to personal data sharing will often exceed its private returns. Using a unique sequence of RCTs for randomized COVID-19 testing among tens of thousands of households in Utah, we analyze different tools to stimulate personal data sharing. We contrast the effectiveness of incentives for data sharing with mechanisms suggested by behavioral economics, including moral engagement, image motivation, and identity. Our results suggest that incentives by themselves can easily backfire and are highly complementary with framing effects. Furthermore, image motivation and identity are an order of magnitude more effective in influencing data sharing than monetary incentives.