The breakaway folks would not have some hidden world akin to a Bond villain or that sci-fi movie about the elites who live in the Niven ring. Essentially they live like our known civilization, they just possess certain technologies a bit ahead of ours, or perpendicular to some of it. In my hypothesis, one breakaway lives among us, they simply have access to things we don't. The other probably lives mostly among us, too, but they keep their thing secret. If the tech is not that far -- or is perpendicular and actually looks like ours -- we wouldn't know it. My opinion is that their most noticeably advanced tech would be space travel technology and that's up in space, not down here for us to constantly run into. Them going back and forth to it could explain many UFO sightings. The very farthest they could have gone, in my opinion, would be a permanent presence on the Moon, maybe Mars.
They might also possess medical tech beyond/outside what we know, likely enhancements on known tech. But for the most part, you're talking about a 'civilization' that operates more like a private club than a completely different civilisation.