Does local storage work across different users
So if I put something in local storage and someone else comes to my code will they see it?
No, localStorage is stored on local machine, not shared on network. So other machines would not able to access localStorage directly.
Calviղ Let's assume I share usage of a laptop. Then I visit a website x, and that website stores data in localstorage. Then my friend visit same website x, with the same laptop and browser. Will the website be using the same data stored in localstorage that was stored when I visited that site in the first place. Maybe the OP was asking that, although I'm not sure XD (Edited)
Ipang Since it's client side data that localStorage stored, it doesn't used for recognizing user normally. Whenever the website is reopened again, it should fetch the same set of localStorage data.
Ipang Programmers still can write code to send localStorage variables to server, the variables is non secured, data like last visited search string, visit counts.
Calviղ Then can we possibly conclude that localstorage is not user specific data (unlike cookies), as per the OP question?
Then how would save stuff cross platform
This is more networking. It depends on how your LAN is set up. The network admin should be the only one with unauthorized access unless there is a specific shared space between all devices on a network. And thats usually manually set up.