Pretty much described what MSFT is trying to do with M365 Copilot. It's "secure", it can be deployed in GCC tenants, etc. It may not be a dedicated private server exactly, but they believe it is private enough.

I wouldn’t put “private” or “secure” in the same product being sold by Microsoft

I don't think there is anything private about that, what I was actually thinking is the polished equivalent of me spinning up LM Studio (or Ollama...) and rolling my own vector database. Mandatory updates and phoning home are a no-go, I just want it to run on my private network the same way my NAS does.