I built a mac .app bundle for one of my games. I can't put savegames inside the bundle, so I put them into /Users/$NAME/Library/Application Support/$GAME/savegames. This has the nice effect that installing new versions of the game with the itch-app will not erase saves games and screenshots. Also you can put the game somewhere read-only for all users and each user has his own writable savegame/options/player profile place. The second point is moot with the itch app of course, as it installs the whole game in a per-user config directory anyway.

The problems are:

1. When you uninstall the game through the itch app, the savegame folder will stay
2. You can't put the downloaded game on a thumb drive

Will itch have a preferred savegame folder outside the game folder? Should I do something different when running under itch? How do other games handle savegames (and screenshots and highscores)?

To answer your last question, for Paper Sorcerer:
I use ORK in Unity, and it's built in that savegames are saved in a preset directory, persistentDataPath:
Windows save files location: C:\Users\username\AppData\LocalLow\UltraRunaway\Paper Sorcerer
Mac save location: Users/username/library/cache/UltraRunaway/Paper Sorcerer
Linux saves: /home/[username]/.config/unity3d/UltraRunaway/Paper Sorcerer

Since it's a file location set by Unity, it isn't possible for me to change the directory manually.

bumping this because of the recent sandboxing feature