Well... it's not quite like that. The major players like ChatGPT allow you to use their engine (their algorithms) and just tailor how it behaves, what it knows, etc. But the core AI engine is still guarded and retained as proprietary, and still retains its base training. (ex: ChatGPT in a private model would still know the difference between a cat and a char, general language interpretation, concepts, etc).
There are some "open source" AI engines that can be more finely tuned. "Open Source" means the actual program code, algorithms, etc, are public, and thus you can download all of it and modify it as you want. But you better know what you're doing.