If I wanted to teach during my postdoc, I would have had to do it an a different institution as an adjunct. Mine wouldn't allow it. Obviously if you are doing a postdoc at a place like a national lab or in industry, you'll definitely have to teach somewhere else.
As far as grants and research direction, again, I'd say it depends. Often times in experimental physics, you are only hired once money is already secured, so you are working on someone else's already submitted grant. Depending on your boss, you might have more or less flexibility about the specifics of the research, and you will probably be expected to contribute to future proposals.
Outside of experimental physics, I would think it could be different. I would imagine doing theory or computation, there's a lot more room to submit your own little side grants, and I've heard that in fields like bio, there is an expectation that you need to support your own salary relatively quickly with your own grants as a postdoc.
So, it depends.