The page MediaWiki:Common.json was introduced in core with change 408940 (linked to T76554). The default text of the page is /* Any JSON here will be loaded for all users on every page load. */. However, I have tested by putting some actual JSON content in my test wiki's MediaWiki:Common.json, and there is no apparent way of accessing what I put there – in other words, it doesn't appear it is loading on every page load. Looking through all the files that were changed in that commit, I can't see anything that actually exposes Common.json content either (I don't know if there are any subsequent commits that have added this, though). None of the major wikis I've checked (mediawikiwiki, metawiki, commonswiki, enwiki) make any use of the page either.
So: How can I use content that I put into Common.json?