If you want your game to support “finding other players” somehow (matchmaking) or “storing online state” somehow (scoreboards, persistent inventory, etc) then you will need some kind of servers for that.
Exactly what kind of servers you need depend on which “online subsystem” you’re using. If you use the Steam online subsystem, and you get greenlit and released onto Steam, you can use the Steam servers for that. If you use Google Play, you can use Google Play servers for that; same things for Apple GameCenter, Xbox Live, etc.
Additionally, if you want to run authoritative game servers (to reduce the chance of cheating, say,) then you need to run those servers yourself; no existing infrastructure will host your game servers for free.
The usual way to get around this is to elect one of the players are “the server” (“host”) for the game, and everyone connects to that player when joining the match.