Hi Cynthia,
Which web app are you using? Are you able to share the link?
Most likely if you are signing in as different users, your network security will not allow other users to access your browsing history. If you are using some sort of open machine, ensure that track your history is turned off if the app is private. Also, if you sign out of your ArcGIS Online account, your session token will be invalidated and users won't be able to access your application through a stored browser cookie.
If a sign out button is essential, please log an idea and let us know the app type.
Thanks,
Kelly
