Managers will often relay signs for things like pickoffs, pitchouts, but also for things like defensive positioning, even when no one is on base. There may be some examples of a manager expressly wanting a particular pitch thrown, but it's exceedingly rare at the Major League level. It's somewhat more common in the minors, where pitchers are on a much stricter regimen and results are not always the goal.
Most college coaches call pitches, though not all. Wayne Graham at Rice famously let/made his catcher and pitcher call their own game, but that is very much the exception. Graham was also quirky about other things such as not having hitters hit off of a tee or soft toss or the usual types of hitting drills. It was always a live arm.
