I am no internet expert, so not sure if and when I'm using the right language.
Let's say we have two web pages (say one is some TiddlyWiki, say the other is a TiddlyWiki with BASIC Anywhere Machine). For them (and however many others) to share data via local storage, they must all be in the same origin. That origin could be any web host at all, as long as all the web pages are from the same IP and port (if I understand correctly?)
If a web browser allows local storage at localhost (i.e. the hard drive on your computer), then all TiddlyWiki instances and whatever other local web pages on your computer are accessing the same local storage.
Something like that ...