A data processing system comprises ownership circuitry to enforce ownership rights of memory regions within a physical memory address space. A given memory region has a given owning process specified from among a plurality of processes and independently of privilege level. The given owning process has rights to control access to the given memory region. The given owning process designates the given memory region as one of: private to the given owning process and shared between the given owning process and at least one further source of memory access requests. A given owning process may deny access to the given memory region to a process having a greater level of privilege than the given owning process. Data stored within the given memory region may be destructively overwritten, and completion of the overwriting may be tracked by overwrite tracking hardware to ensure completion of the overwriting before the new owner obtains rights to control access.