There is a method GetOwnedGames (https://developer.valvesoftware.com/wiki/Steam_Web_API#GetOwnedGames_.28v0001.29)
but it does not work if the user has set up a private profile.
Are there any other ways or methods to find out that the user owns our game?
The colleague has found more complete documentation of the api steam methods. There is a method for obtaining information about whether the user owns our game. https://lab.xpaw.me/steam_api_documentation.html#ISteamUser_GetPublisherAppOwnership_v1