Long Version
If you look at any storygame, you'll see the first part of the URL is http://chooseyourstory.com/story/ and this tells you that the link is for a storygame.
If a story is not on sneak preview, you have to be an author or coauthor to be able to access the link (where the link is http://chooseyourstory.com/story/[storyname]. Note that the story name, if it contains special characters, will have to be parsed. i.e. Your story is called 'RavenFall - Demo', but the parsed variant (which is the part at the end of the link) is 'RavenFall%20~2d%20Demo'.
If a story IS on sneak preview, then anyone can access the link, and the storygame will be listed under 'Storygames' on the author's profiles, but will not be hyperlinked (as this is reserved for published storygames).
However, the link, in this case http://chooseyourstory.com/story/RaveFall%20~2d%20Demo, is still accessible by anyone, but slightly annoying if the hyperlink isn't provided (as you've got to change it yourself). Luckily this is still relatively easy, for spaces are automatically changed, and other special symbols can be found easily enough by looking for a published storygame with the desired character in its title, or if this fails, making a storygame yourself with the desired symbol.
Note that the CYS extension provides links to the storygame eliminating this hassle (as far as I'm aware).
So, assuming I interpreted this part correctly: 'My only problem is, I can't find a way to privatize it so that only people I link to it can read it', then you are out of luck as what you are looking for doesn't exist. The story is either on sneak preview (available access to all) or not (only available to authors). Further, I think there was a limit of five for coauthors.
Anyway, I guess if there was a way to hide the sneak preview story from the profile page you could get what you want, in that the link is the only way to get to the story, and that there is no way to get the link (as there is no story name anywhere for people to figure it out).
I will mention that people generally won't bother with sneak preview stories if they don't know the author or if the author isn't asking people to look at it, so I imagine this still works as a solution, but I did want to make sure you knew it is easy to access sneak preview stories even if an author doesn't give you a direct link.
Short Version
If a story is on sneak preview, it is easy enough to access without the link being provided by the author, so it may not be as private as you are looking for (assuming I understood this properly: 'I can't find a way to privatize it so that only people I link to it can read it', as it won't ONLY be people who you link that can read it, but anyone who can write up the link).
Game Script
You didn't get too specific about what precisely you mean, but the phrasing made me worried that you'll be trying to make a game game as opposed to a storygame, and I wanted to warn you that tools for storygames are not a game engine.
While you can make certain styles of games, or more game focused storygames, making works which almost look like proper games is something that, while possible, is extremely tedious, and the amount of effort that'd go into working around the limitations would probably be a bit pointless, considering there are better tools out there for game creation.
Granted, I'm no expert, so it might not be as difficult as I'm envisioning, but a warning, just in case, won't hurt anyone.
Conclusion
Best of luck with your project.