Land and houses aren't private property unless you're renting them out. If they aren't a financial asset, they're just personal property.

Businesses are an interesting question? The Federation, or at least its core worlds, doesn't use money (by the 24th century). The only business we see onscreen, on a Federation core world, as far as I can remember, is Sisko's Creole Kitchen. If there's no money, why does Joseph Sisko run it? My guess is to maintain the tradition of Creole cuisine, to perfect his skills as a chef, to meet and interact with guests, and to preserve an historic New Orleans building by keeping it in use. Is it private property? Does he own it? He owns the business in some abstract sense, but the building? Probably not. I'd expect he holds it in trust in some kind of legal arrangement with the city, but there's really no onscreen evidence.