I compare a dogs mouth to a toddler's hands and mouth. You can't keep a thing out of it. You have to take it out, teach leave it and still - the lure of that stick seems so great that they will try to step over the line if allowed.
As humans - sometimes we can't help just allowing as its so cute but watch out - when you do, they may take a mile. Be sure you can live with it. A stick my dissolve in most cases but not necessarily and the damage it can do to bowels - well we won't go there so practice - 'leave it' or 'drop it' and see how many times you have to practice to get there.