If you try and enable the screen sharing feature during a call on Hangouts on Chromium on Debian, it happens absolutely nothing, no error is reported.
This is because the Chromium package in the debian repositories is built with the enable_hangout_services_extension=false flag (every other feature of Hangouts actually works).
You can download the source, edit the rules file and enable that flag. The produced Chromium package will have the screen sharing feature correctly working.
Check here to see the detailed build instructions or to download the ready binary.
Another option is using Firefox ESR.
Context: debian Sid - August 31, 2017 - chromium_60.0.3112.78-1