Non-HTTP requests inside own OSM server

You don't have to use HTTP for overpass api on your own server. There's a binary called osm3s_query which will happily accept your Overpass XML or QL queries on stdin and return some OSM xml result on stdout. That's actually quite handy once in a while for some local tests. Even overpass-api.de leverages osm3s_query for the purpose of creating/updating areas (see Wiki for details).

However, HTTP is clearly more convenient for overpass turbo and other web based apps, even if you're only pointing your browser to localhost.

This brings up the question why you want to avoid HTTP in the first place??