Local does a pretty good job of scrubbing private info from the logs and the errors it produces, however there’s always the possibility that something private can come through. Because these are public forums, always review the screenshots you are sharing to make sure there isn’t private info like passwords being displayed.
To clarify - are you saying when you close Local the site is no longer accessible?
If so, that is expected behavior. Once the Local app is closed, the site will no longer be available. In order to view the site the Local app needs to be open and the site needs to be in the “Started” state.
When I stop the website in local I cannot access the LIVE website using Firefox. I have to delete the entry made by local in my hosts file to be able to access the live website.
Did you happen to set the Site Domain in the Local App to the exact same as the Live website? If so, your computer will still point to the Local site.
Are you able to access the live site from another device? That would be another clue that your computer is pointing to the local version of the site when hitting that domain.
I would recommend updating the Site Domain in the Local App to something else (like yourdomain.local, etc). You may want to make a backup (export) of your site before doing that.
Here’s a help doc walking through how to export your site:
I tried that but then it breaks my plugins that require authentication like elementor pro for example. I can’t develop the websites without certain plug-in functionality.