There are several variations of Spectre. The first variant ("Spectre V1") lets a process read its own memory; the second variant ("Spectre V2") lets a process read another process's memory.
Web browser manufacturers seem to be focused on preventing Spectre V1, although I'm unclear on whether that's because V2 is too hard to exploit from JavaScript, is mitigated in other ways (e.g., CPU updates), etc.