Open your story, then select "Publish to file" in the bottom left menu. Then share the produced .html file by whatever way you prefer. The person who receive that file can play it by opening it in a browser, or can import it in Twine 2.0 to edit it.
I wanted to add a little bit of background to narF's answer to help clarify.
Everything you edit in Twine is stored on your own computer only, even if you're using the version at twinery.org/2. Your story is not stored in any way on a server. That's why it's super important to regularly save archives of your stories with the Archive button on the right side of the story list.
Because of this, the only way to get a shareable version of your story to give to someone else is to save it to a file, as narF explained. They can either import it into their Twine editor and make changes there, or just view it in their web browser. The same file that is playable in a browser is also importable in the editor.
Hope this helps! I realize this is an unusual way of working, if you using the editor at twinery.org/2.
I would also suggest to anyone sharing stories via the Publish to File featues that you add a version number (x.x.x) or a timestamp (YYMMDDHHmmSS) to your story content as well as to the file name or you will find it is very easy to confuse which version of the story people are looking at. It is especially important if more that one person is editing the contents of the story.
In my experience the importer does not know or care if the version your about to import is older or newer than the one in currently in local storage.
Comments
Everything you edit in Twine is stored on your own computer only, even if you're using the version at twinery.org/2. Your story is not stored in any way on a server. That's why it's super important to regularly save archives of your stories with the Archive button on the right side of the story list.
Because of this, the only way to get a shareable version of your story to give to someone else is to save it to a file, as narF explained. They can either import it into their Twine editor and make changes there, or just view it in their web browser. The same file that is playable in a browser is also importable in the editor.
Hope this helps! I realize this is an unusual way of working, if you using the editor at twinery.org/2.
In my experience the importer does not know or care if the version your about to import is older or newer than the one in currently in local storage.