How to properly decouple Django from an AJAX application?
A mobile client doesn't care if the Javascript comes from Django or any other web server.  So go ahead and put all your JavaScript and static HTML on another server.
If you want your mobile app to see if the user is logged in, it should make an AJAX call to your Django backend (where the request is authenticated).  The data returned should indicate if the session is active (user is logged in).
Another AJAX call can perform the Django logout function.
I'm using TastyPie and Django to build out my backend for an application that will have browser and mobile (native iOS) clients.
I have been through the TastyPie and Django docs, can authenticate successfully either using the TastyPie resources I set up, or using Djangos built in views. I see a lot of examples on including the CSRF token on the page and grabbing it with your JavaScript, and that works, but I don't understand now to actually determine whether a user is logged in on initial page load (from JavaScript). 
Example:
If I want to serve static HTML from a separate, fast web server, and cache my application JavaScript, and only interact with Django through TastyPie views, how do I determine if the user is logged in (and know to render a login form or the app views using JavaScript), and after logout, is there any session information I need to remove from the client browser?
If I were to serve up HTML through Django's template engine, I could render the login form through there appropriately, but that seems not ideal if I want to truly decouple my JavaScript app from Django (and behave like a mobile client).
Edit: I am using Backbone.js, but I don't think that should matter.
UPDATE:
I think I figured it out reading through Django's CSRF documentation again.
If your view is not rendering a template containing the csrf_token template tag, Django might not set the CSRF token cookie. This is common in cases where forms are dynamically added to the page. To address this case, Django provides a view decorator which forces setting of the cookie: ensure_csrf_cookie().
If I do not want to render Django templates, this reads like I can still use the cookie and pull that into my Backbone or jQuery AJAX methods. I'm not sure if TastyPie ensures the cookie will be sent or how to tie into it.
If I use AJAX to logout, will the cookie automatically be removed or does it become invalid? Are these CSRF tokens unique to each user session? I'll have to test some things tomorrow with it. Is it possible to use Django decorators on TastyPie views?