If you want client access to some data that is only available on the server, then you need to send an ajax call from the client to your server and you need to create a route on your server to respond to that ajax call.
Client code runs only on the client and has no direct access to any data on the server.
Server code runs only on the server and has no direct access to any data on the client.
To communicate between the two, you have to send a request from one to the other and then return a response. Usually this is done with an Ajax call sent from client to server. You could also establish a webSocket connection between the two and then either client or server could send data to the other.
The server also has the opportunity, when creating the original page content, to embed settings or values in the page itself, either as Javascript variables, as HTML values or even as a cookie. This obviously has to be done ahead of time when the page is rendered so it can't be a request that the client comes up with later after the page has been rendered to the client.
FYI, in the particular example you show, it is common for a client to be able to tell if it is logged in via some state in the page (either the presence of a particular cookie) or something else embedded in the page by the server. This isn't necessarily secure and isn't the way the server would tell if a request was logged in, but it usually suffices for client-side logic to decide how it wants to behave.