In Rovinj (Istria), at least some of the local tourist agencies had books of sobe listings. I did not see any private parties hanging out around the bus station and offering rooms in either Rovinj or Zagreb. I did not go to any islands on my last trip to Croatia.
I believe I did see some "sobe" signs displayed on houses. Don't forget to check on the air conditioning situation if you're traveling during the summer.
Booking.com has some rather informal listings. I ended up in two efficiency apartments and one room, all found through booking.com. None was in somebody's private home; the latter happened only in Rovinj, via a travel agency.
One thing to keep in mind is that smoking is considerably more common in Croatia than in the US. Most if not all hotels have rooms that are non-smoking, at least in theory. (I think there's an EU regulation.) In someone's private home, the conditions can be quite different. I speak from experience when I say that you may be dealing with active second-hand smoke, not just a lingering odor.