Does the side without a lockring have left-hand threads showing beyond the cog?
If yes, you have a fixed-fixed hub. You can get another lockring for it. (Be aware, there are two lockring standards. A few Italian hubs use a larger diameter lockring. Miche does on a lot of their fix gear hubs but not all. I would unscrew your current lockring and very gently start the replacement onto it to check (no tools, fingers only,)
If no, ie the threads you can see are right-hand, then you have the far more common fix-free hub where the no-ring side is intended to have a singlespeed freewheel. The fix gear cog will work without issue going forward but if you ever try to brake, intentionally or no, you might unscrew the cog, leaving you in "neutral" and probably throwing your chain which can lock up the rear wheel. If you are at speed, this may well destroy your tire and will scrape paint, gouge the chainstay and quite likely bend a few chain links. It's also quite exciting.
It is also possible the side without a lockring has damaged threads and can no longer accept one. if so, consider this a compromised hub.
BTDT on all of the above scenarios. Rode Cycle Oregon last summer on a hub with stripped lockring threads on both sides. Had good, working brakes on both wheels and rode up, around and down from the Crater Lake rim without issue. (But next time, it's going to be on a hub that is "right".)
Ben