The process is roughly the same as inspecting the UI, you basically enable remote debugging and then launch the Browser Toolbox: https://superuser.com/questions/1608096/how-to-inspect-firef...
(this is also a great way to figure out what you need to do for a user-chrome CSS file, e.g. if you want to change how tabs appear beyond just theming. though using a user-chrome CSS file is unfortunately a minor pain in the ass.)
I had to switch it to "multi-process mode" to get network requests from extensions to show up, but that broadly makes sense.
e.g. here's ublock pulling a random block list that I clicked on to test: https://i.imgur.com/wBYbBM2.png
and here's Firefox itself updating the safe-browsing list in the background: https://i.imgur.com/JYxWwjW.png
A well made extension will only pop up an occasional toast. One that needs the server to work will show the toast when the user expects it to be talking to the server. One sending keypresses or other suspicious data will be really obvious.
For example, even if the extension itself can't make requests, there are plenty of ways to 'trick' a webpage into making a request on your behalf when you have permissions to modify the Dom of the web page.