They are many ways or methods to write web services and to handle sessions (logged in/logged out states). Hence, its not easy to answer without knowing how your service works.
For an example, Session managing techniques.
Browsers handle sessions in a different way. In apps, we will have to handle them manually.
First when you open your service in browser(ex chrome), open developer tools (shortcut - F12) go to Network
sectionand observe each request and response and what headers are set. That gives you some idea. Then come back here and update your question with more details.
Also, you can refer to this SO question on session management