Unfortunately to protect user privacy and to avoid potential cross-site
scripting attacks the JSON-P endpoints you're using have no concept of
"the current user". In order to handle user logins it is necessary to
implement some server-side logic to perform an OAuth transaction.
The OAuth flow will include a step where the user is asked to grant your
application access to his or her account, after which it is possible to
identify the user using some combination of the techniques described on
this page:
http://www.typepad.com/services/apidocs/authentication
Regrettably we do not have a solution that's available to purely
client-side applications at this time. Applications running entirely in
the browser, such as widgets, may only access public data that does not
require user authentication.