The Story Projects you create using the Twine 2.x application are only stored locally on your machine, and nothing is stored on the (web) server even if you are using the web-browser based release that is hosted on the Twinery web-site. Where exactly on your machine those projects are stored depends on which release of the application you are using.
1. The Web-browser based releases (both the hosted copy and the downloadable copy)
These releases store your projects within the Local Storage cache of the web-browser you are using to run the Twine 2.x application.
2. The Desktop application based release.
This release stores your projects within a folder on your local hard-drive, the exact location of this folder can be determined by using the Twine > Show Library menu items.
The Story HTML files created via the Publish to File option are also stored locally to your machine, generally in a folder of your choosing. The Twine 2.x application has no built-in ability to publish your stories on-line, that activity is left up to you to do manually.