The answer is simple and complicated. There isn't an absolute answer, but there is a pattern. Usually, the offensive coordinator calls the plays. Not always though and you would never say "never" when asked if the quarterback is allowed to change the play.
The Browns run a great offense. I love coordinator Bruce Arians' offense.
He calls most of the plays on game day, but Kelly Holcomb and Tim Couch have lots of flexibility. Holcomb's work in the Indianapolis game is a good example of the freedom Arians gives his quarterbacks. Many times, early in that game, the Colts would come out showing blitz and off coverage. Kelly would audible. Then the Colts' defense would audible into a "cover two." Holcomb would then audible off their audible. That's why the Browns were able to complete so many passes down the middle that day.
The game becomes a big chess match and the goal is to move the football and score points. It sounds simply obvious right? Not necessarily.
In college, I was fortunate to have Howard Schnellenberger as my head coach. Coach Schnellenberger told me "you'll never play quarterback if you don't change plays." He would always remind me that the head coach can only guess what the defense is going to do, but the quarterback, "now, he's the last person that sees the safeties," he'd say. The quarterback can see how the defense is lining up over his offense. "You should be changing plays," Schnellen-berger would say, "because you can see it all in front of you." Even as a red shirt freshman in the 1984 Orange Bowl, I was changing plays, with Coach Schnellenberger's blessing. Nebraska wasn't disguising their coverages that night and I'd say I changed 70% of the plays we ran in the Orange Bowl game. We scored over 30 points and won the National Championship.
It's not about insubordination or glamour-seeking. It's just about the right way of calling plays.
My second year in the NFL, Marty Schottenheimer was the head coach of the Browns and Lindy Infante was the offensive coordinator. Marty and Lindy didn't want the quarterbacks to change the plays. Gary Danielson fought tooth and nail to get them to give us some leeway. As time passed, Lindy gave me an awful lot of leeway. The play Lindy would call into the huddle became more of a suggestion than anything else. Except when the football was inside the 10-yard line. Then we stuck with the play he called. I've always felt that I had a really good feel for changing plays, especially late in my career when I felt I was really under the microscope.
Bill Belichick never wanted me to audible. He wanted his quarterback to stick with the play sent in from the sideline. No questions. No alternatives.
I clearly remember two times when I changed his play. Both times ended with positive results but negative feelings. In one game, we were playing the Miami Dolphins at the Stadium. Now, I've always looked at football as a game of mismatches. Michael Jackson was one of our wide receivers that year. He's 6-foot-3. Belichick told me before the game not to do any audibles. At one point during the game, we came out of the huddle, lined up and I could see that a 5-foot-8 cornerback was covering Jackson.
Belichick saw the same thing and he was yelling at me from the sidelines not to audible. He knew what I was thinking. He saw the same thing that I saw. A mismatch! Still, he called out for no audible. Guess what? I changed the play. I sent Michael Jackson on a fade route and hit him for a touchdown. Oh, did I get yelled at on the sideline.
Yes, it's true; this next play was drawn up in the dirt of the stadium. We were playing the Denver Broncos midway through the 1993 season. The Broncos had two aggressive safeties, Steve Atwater and Dennis Smith. In the huddle, I told the tight end to stay tight and kept two backs in the backfield. I had five linemen and eight guys total ready to block. I told Michael Jackson to run a route 17 yards up, cut two yards in on a square, I'd pump fake and when the safeties came up, he was to go deep. The play worked. It went for a touchdown. Ironically, that was my last pass as a Brown.
Belichick cut me the next day.
When I played for the Dolphins, offensive coordinator Gary Stevens would call all the plays except when we got into short yardage and goal line situations. Then, it was head coach Don Shula's call. Shula looked at those plays as being so critical and important, he wouldn't allow anyone else the responsibility.
Really, that's the way it is in the NFL today. There's an awful lot of pressure on head coaches. Because of that pressure, coaches want to be in control of their own destiny. We're really getting away from the quarterbacks making their own calls and changes.
Unless you're an established, fortunate or opinionated quarterback, it's not going to happen. I can count the number of quarterbacks allowed that luxury today on one hand. Peyton Manning is one of them. Why? Manning gets it AND he has shown the ability to consistently produce when given the freedom.
The reasons to give him reign of an offense speak for themselves.
Is the trend of quarterbacks being allowed less and less permission to deviate a good one? In some cases, yes. Take Baltimore starting quarterback Kyle Boller as an example. You have to have a coach in control of Boller. He's too young. In Manning's case, no.
You've got to keep people off balance. Defenses are too good and the players are too talented today. Very few teams can run when the defense knows you're going to run. The same can be said for the pass.
So, what's the bottom line? It's simple really. Nothing beats having great players. No matter who's making the calls.