Over the past few years, calls for greater transparency and accountability in the natural and social sciences have intensified. This push has been driven in part by controversies among scientists over the integrity of their research methods, conflicts of interest, the inability to replicate the findings of well-known studies, and by the failure to share or publish experiments that do not turn out as expected. But it is also driven by a desire to protect scientists from attacks by non-scientists defending certain economic interests or political agenda. The emergence of the Open Science movement coincides with a vast increase in the volume and accessibility of data on individuals and the growing power of big data analytics. These trends raise significant concerns about how to continue protecting the privacy of people who participate in social science research while being open and transparent about data and methods. This paradox is on full display in the recent decision by the US Environmental Protection Agency to prohibit evidentiary uses of scientific studies without full disclosure of identifying information on all participants to allow for independent verification of the results. The presentation includes an overview of how the data sharing/data privacy paradox emerged and a look at trends in public opinion on science.