https://switch2osm.org/serving-tiles/building-a-tile-server-...
There certainly isn't much documentation about setting up a private editing API, but there is quite a bit of information published in this repo:
The reason was to make myself more independent of web services.
If you aren't the nexus of worldwide OpenStreetMap editing (which parts of osm.org are that), there isn't so much reason to run a whole bunch of the stuff. You don't need to support thousands of people making edits, if you aren't capturing edits you don't need a system to replicate them out to data consumers. Etc, etc.
You wont have any of the history, and no users info etc, but you can edit and if you have tile server set up can serve tiles from it as well.
Far better though is to import the data into PostGIS (the rendering database) and edit and add that way with a desktop gis.