OSM tile server, combination of my own private data and OpenStreetMap priodic updates

I have set up an OSM tile server and also an OverpassAPI on my own server. While I was doing that, I understood that OpenStreetMap provide minutely, hourly and daily updates. So my servers can be sync with it.

On the other hand, I want to add my own private data into my server without pushing them to OpenStreetMap servers.

My question is that can I have a tile server(or OverpassAPI server) that feed from both sources(my own private data and periodic updates of OpenStreetMap)?

Any assistance would be grealty appreciated.

Note: Technically speaking, I am really confused. I have couple of questions but the most one is that I don't have any idea about ids. I know that OpenStreetMap data(nodes, ways and relations) have positive ids and therefore my own data could get negative. But is it a real solution?

This is a recap of my question in StackOverflow(here). There are also some similar question(i.e. this,this). None of them have accepted answer.