I am not sure what you are trying to accomplish. I think you might want to clarify where you refer to host threads and where to GPU threads.
Any auto variable in device code is local to a GPU thread: it gets mapped to either registers or to local memory, where the word “local” in the term “local memory” means GPU-thread-local. So local memory is a thread-specific mapping of a section of global memory.
Since textures are read-only data objects, I am not sure what influences between the threads in the same process you are concerned about. By definition, host-side threads share most of the resources (including memory) owned by the host-side process to which they belong, with the exception of registers and thread-local memory.