Chat users still logged on even when they are actually offline

If users are unable to find a solution through the Chat steps concerning offline activity refer to your "Security Settings" for your overall org and enable your session timeout settings. 

Classic
 

1. Go to your "Security Controls" drop down and select "Session Settings."
2. Under "Session Timeout" section, users can select the time out value of when you would like the user to be logged off. The time ranges from 15 minutes to 12 hours.
4. Choose the time you would prefer and select the "Force logout on session timeout button" box. Save what you have enabled.   
 

Lightning

1. Go to Setup.
2. On Quick Find Box, search for "Session Settings."
3. From the "Timeout Value", users can select the time out value of when you would like the user to be logged off (The time ranges from 15 minutes to 24 hours).
4. Select the "Force logout on session timeout" box.
5. Click Save.

A session timeout has been enabled. This will make it so that if there is inactivity for the allotted time you choose on Chat then the session itself will time out and the user will be logged off.