I haven't had the guts to try any screwdrivers or allen keys with my looser back handle rings. My LD is preloved and Dior will not repair without proof of purchase, even if I'm willing to pay for the service. Keep in mind this is a 6 year old bag that has been well worn/loved so I'm not surprised the handles have become loose over time. What has helped is wrapping some thread tightly around one of the rings (on the inside) so that it sits in the tiny grove - this reduces the side to side swinging somewhat. The ends of the thread are visible if you look closely but no one is going to inspect the inside of my bag. The thread can easily be removed as well, so that's a plus.
If you have a LD handle that has become a little loose, it's worth giving this method a shot if you can't get Dior to repair it for you.