Prospects have never seemed better for a truly global approach to science to improve human health, with leaders of national initiatives laying out their vision of a worldwide network of related projects. An extensive literature addresses obstacles to global genomic data sharing, yet a series of public polls suggests that the scientific community may be overlooking a significant barrier: potential public resistance to data sharing across national borders. In several large United States surveys, university researchers in other countries were deemed the least acceptable group of data users, and a just-completed US survey found a marked increase in privacy and security concerns related to data access by non-US researchers. Furthermore, diminished support for sharing beyond national borders is not unique to the US, although the limited data from outside the US suggest variation across countries as well as demographic groups. Possible sources of resistance include apprehension about privacy and security protections. Strategies for building public support include making the affirmative case for global data sharing, addressing privacy, security, and other legitimate concerns, and investigating public concerns in greater depth.