So this issue happened with my iPhone 6S twice, first after I got my phone (erasing and restoring from iCloud fixed it), then a second time later. After the second time it happened, I realized that, unless I'm completely mistaken, that it happened when I turned off my phone (not a hard restart). When I first got my phone at the end of December, before I finished setting everything up, Siri worked fine, but after I restarted my phone to transfer over a number, I noticed Siri stopped working. As I said before, restoring fixed it. However, later I turned off my phone before I went into an exam as an extra precaution, and I noticed when I went to use Siri the next day that it had the same problems as before. I don't know if powering off my phone really did cause those issues, or if that is when it started for anyone else, but Siri was working fine both times before I shut my phone off.
Edit: And if that is true, my concern is that a future iOS version, while bringing back Siri for the moment, will still have the issue come back after restating my phone again sometime later.