There is a plugin for this: Join My Multisite
Allow site admins to automatically add existing users to their site,
or let users decide at the click a button.
I don’t know how important it is to you that the users don’t know that they are in a multisite but the list of the plugin’s options is very interesting:
By activating this plugin, you give your Site Admins the following
options:
- Auto-add users
- Have a ‘Join This Site’ button in a widget
- Keep things exactly as they are
- Create a per site registration page
- Use a shortcode to put a ‘join this site’ button on any page/post.