Rc and Arc are about (shared) ownership not borrowing. If they're turning off anything, it's ownership. Borrowing works the same way as ever.
In fact that's an advantage of Rust here: because of the borrow checker you can safely get a reference to an Rc's contents and hand it off to something without having to alter the refcount, so you get significantly less refcount traffic than in other languages where such a thing is unsafe (or not under your control). That's especially important for Arc.
It also provides for nice safe optimisations like "move out of this Rc if I'm the only owner" (Rc/Arc::try_unwrap).
In fact that's an advantage of Rust here: because of the borrow checker you can safely get a reference to an Rc's contents and hand it off to something without having to alter the refcount, so you get significantly less refcount traffic than in other languages where such a thing is unsafe (or not under your control). That's especially important for Arc.
It also provides for nice safe optimisations like "move out of this Rc if I'm the only owner" (Rc/Arc::try_unwrap).