I have managed to use oauth authentication and add a Sign in with Twitter functionality to a Google App Engine web app. 
How should I verify, during the site navigation, if the user is still logged in Twitter?
Are you talking about being logged in to Twitter.com or your app? If you have received oAuth access tokens by authenticating an app, then logging out of twitter.com won't 'log you out' of any apps, the tokens will remain valid until the user revokes the access.
Sign in with Twitter: how to verify the current user is stil logged in