Puzzlescript games accessed with links like https://www.puzzlescript.net/play.html?p=4b0af10c54188fc97599092f2683fea7 are stored on GitHub as gists and fetched from there with their Developer API. Their privacy policy is here.
Why store things there? Because they have to be stored somewhere, the sharing functionality is really important for PuzzleScript, and it makes creating and sharing games really easy. And GitHub takes care of moderation/spam filtering, which otherwise could cause a lot of stress.
If you don't want to use the online-sharing facilities you can still save/load game locally, and can still export standalone html files.
Puzzlescript Next can save things to your browser's local cache - it's just used for loading/saving files in the file save dropdown. Then to create a shareable link to your game you will need to authorise with your own GitHub account. Then, when you click on share in future, a publicly-accessible copy of your game is uploaded to a gist in your account.
Puzzlescript Next