Yes you can? Sure you can't run GPT5 locally, but get your hands on a proper GPU and you can run some still very sophisticated local inference.

You can do some, but many of them have license restrictions that prevent you from using them in certain ways. I can buy an Intel chip and deliberately use it to do things that hurt Intel's business (e.g., start a competing company). The big AI companies are trying very hard to make that kind of thing impossible by imposing constraints on the allowed uses of their models.