When researchers talk about "sharing" datasets, they generally mean publishing the data so anyone can access it - not just sharing access to the data when asked. There are many reasons to make a dataset publicly available, such as:
Important factors to consider when choosing a repository are security (will the data become corrupted or be manipulated?), length of access (will my data suddenly disappear?), and access (will researchers be able to find my dataset?). Alternatively, the Eunice Kennedy Shriver National Institute of Child Health & Human Development (NICHD), Federally Funded Research & Development Centers, and MIT Lincoln Laboratory have created a Data Repository Finder. The NIH also has further guidelines on selecting repositories for data.