Techniques are described for managing access of executing programs to non-local block data storage. In some situations, a block data storage service uses multiple server storage systems to reliably store copies of network-accessible block data storage volumes that may be used by programs executing on other physical computing systems, and snapshot copies of some volumes may also be stored (e.g., on remote archival storage systems). A group of multiple server block data storage systems that store block data volumes may in some situations be co-located at a data center, and programs that use volumes stored there may execute on other computing systems at that data center, while the archival storage systems may be located outside the data center. The snapshot copies of volumes may be used in various ways, including to allow users to obtain their own copies of other users' volumes (e.g., for a fee).