Cookie is always carrying something in her mouth (like all retrievers...) . We've got a large collection of sticks in the front yard that she can bring along, but she learned to drop them when we opened the gate to our house.
I let her carry around anything she feels like when we're outside, but she's not allowed to eat food she finds. She has to drop it and I throw it away.

From what I understand, there's no harm in shredding sticks. Most dogs do it.

I think it's good to teach your dog a command that works well for both of you. 'Drop' is a good one, that's what we use. As with everything else we teach our dogs: it takes time and patience!

I know a Golden that chewed on stones as a pup and the owners didn't really get her to stop. That Golden has a few stubs in her mouth now. Very sad.