Have you tried discussing with him how his talking affects the other people in his class? You and I will instantly recognize that being finished with something does not entitle us to distract others who are still working on their task, but a third grader may not think of it.
Just having an arbitrary rule sometimes is less effective. If he is generally a polite, thoughtful young man, then remind him that refraining from talking while the other students are still working is good manners. It is rude and unkind to disrupt students who are still working. It may give him a different way to think that helps him control it.
Also, while getting his stick moved once a week or so isn't something you want to see, it really is all that is needed, if you ask me. Do you really expect your child to be perfect? Of course not. And so he will sometimes get his stick moved due to his weakness--talking. His teacher suggested working on it, and you are. But there is no need to make a bigger deal out of it, than it is.
I wouldn't punish him at home for it. However, if you'd like to implement some sort of positive reinforcement, that is something that would make sense. And extra at home, for extra effort at school...
If he goes a week without his stick being moved, then he can _______. If his stick is moved... no punishment. But no bonus, either. Just be careful that you don't end up with a situation where if he talks on Monday, he says, "oh well, the rest of the week won't matter." or something. LOL
You might ask the teacher if she would consider using a "code" that she can use to remind him to be quiet, without having to actually call him out. But, really, it seems like that is what the stick being moved is all about. There is no punishment attached to the stick being moved, right? It is a warning.
If he is getting it moved multiple times a day, then you have an issue that needs more serious attention, but that doesn't sound like what you have described.
Good luck.