If you're already making calls to server side code with ajax why not just create your own method that you can call from ajax to check the user?
This thread on the asp.net forums may help.
Using hidden fields could also be done, but just be aware that client side things could be modified so any server side code that does things on behalf of the user should also do their own checks to validate that the user is authenticated.
Context.User.Identity.Name gets the user name from the authentication cookie which will either be the email or login name depending on which you are using to login. SiteUtils.GetCurrentSiteUser() is looking up the user based on Context.Identity.Name from the auth cookie, it returns null if the user is not authenticated.
Hope that helps,