I think there's a misunderstanding of what's being discussed in the comment chain. A "private game" doesn't mean anything besides that it's private, such as with a password or a whitelist. It's not the opposite of a dedicated server, nor has anything to do with whether or not you port forward. Anyone can make a private server.
It's not the opposite of a dedicated server, which could more accurately be described as a listen server. But when people think of games like Don't Starve, Garry's Mod, and Terraria, those are actually using Steamwork servers.
When Turkey-er mentioned P2P, they're probably referring to peer-to-peer (not pay-to-play, which I think you may have interpreted it as).