Jackson County is riddled with hidden gem waterfalls. Ours just happens to be readily accessible and went into the state park system. Prior to that we did not post or restrict access and there was a roadside pathway away from the houses. It's not a twin Falls though and is only about 80 foot high at the overflow. The surrounding bluffs are quite high, pushing 300 feet prob in some places. My niece and some of her friends used to trek to alot of the type falls in Jackson County you are referrring to. Most don't mind if you take care of your visit. i'm not aware specifically of which one you are referring to. Jackson County is fantastic for outdoor lovers to get out and explore these things. Falls and vistas are abundant. You can be driving north on the highway that goes through Dodson Branch through rolling pasture and in an instant you are on high ridges that go down to roaring river and back to Gainesboro. I'm more of a transplant so I don't know the crooks and crannies. Dad was from here, but I was born and raised in GA when he worked for UGA. I can walk across road to our small field and look back over Cookeville for miles. And can walk 60 feet behind the house to the bluff and view our falls. I spent more time at the falls visiting than I ever have living on top of it. Funny how that works.
You should also look for Virgin Falls area in White County. Mile or so hump in. Between Sparta and Pikeville. Then you get to a small, wide falls about 20 feet high that lands in a disappearing pool. The water comes up from a spring at the top and only travels a very short distance. We would drink right from the source where it came up. Good day hike outing. I've heard Standing Stone up around Hilham in North Putnam, maybe Overton is quite nice. Never been. Putnam's topography is also diverse. I'm sure alot of this spills over to them. Especially on the east and north areas.
Edit: You got any idea which area of the the counties you might have been on those twin falls?