The pitcher has final say, but a good battery (pitcher-catcher combo) is especially valuable because it lets the pitcher focus on throwing his pitches (and tracking runners as needed) instead of also involving him in the batter-pitcher mind games. Smart catchers are very under-rated by most casual fans, and can tell you why they called every single pitch in every single at-bat in a ball game (which will run well over a hundred pitches a game).

The pitcher does get to choose what to throw. The catcher signs, then the pitcher either shakes his head "no" or winds up for the pitch. If "no", the catcher chooses another pitch until they agree.

The pitcher absolutely does get to decide what pitch to throw. If the pitcher does not like what the catcher puts down he just shakes his head and the catcher puts down a different sign. Repeat until the pitcher is happy.

The big reason why pitchers don't signal to the catcher is because pitcher and catcher must both know what pitch is being thrown beforehand and the batter can see the pitcher but not the catcher, so it's 1000% easier to keep the pitch secret when the catcher calls the pitches.

However, the catcher calling the pitches is beneficial because it allows the pitcher to better concentrate on the act of pitching.