Oh! I actually found the answer here:
https://github.com/itchio/itch.io/issues/1498
You have to make a GET request with the following structure:
GET /games/12345/ownership
Authorization: Bearer <user's oauth token>
Response: { "owns": true, "game_id": 12345 }
In a curl command it would look like this:
curl -X GET "https://api.itch.io/games/GAME_ID/ownership" -H "Authorization: Bearer OAUTH_ACCESS_TOKEN"
Replacing GAME_ID with the id of your game, and OAUTH_ACCESS_TOKEN with the token your user got after authenticating.