I would agree, based upon my reading, that if there is an improvement in Dark Circles it will be via Liver and Kidney cleansing. I may have also read something about the thyroid, but I'd have to reread a bunch of stuff to know for sure.
One of the side effects I found with Oil Pulling was a lot of wrinkles around my eyes disappeared, and believe my skin is much more elastic. I've worn glasses for over 50 years, so most of the skin damage around my eyes is related to sun magnification. My eyes are darker directly under the lids, but again, I think this has to do more with discoloration related to UV rays being heated by the glasses.
Good luck, I think you'll enjoy OPing. If anything else my mouth feels super clean.