Can you explain more? My understanding (which seems to be backed up by the implementation) is that I can call initstate_r in each thread to initialize thread-local state, then call random_r without synchronization from each thread as long as I used that thread's thread-local state.

random_r's internal state is all, from what I can see of the implementation, entirely passed in by the caller. Implementation here: https://sourceware.org/git/?p=glibc.git;a=blob;f=stdlib/rand...