Importance to Goalkeepers
Hand distribution allows a goalkeeper to re-enter a ball into play once they've caught it. There are a lot of different methods to use depending on the distance and accuracy needed, and the positioning of your teammates and opposition. Since the opposition isn't allowed to knock the ball out of the goalkeeper's hands, this is a great opportunity to change the pace of play by counterattacking quickly while the opposition is out of position or slow down the pace and let your team get in position.
Hand Distribution Rules
There are three important rules in hand distribution.
The goalkeeper can only have the ball in their hands within their 18-yard box
The goalkeeper can only hold the ball in their hands for a maximum of 6 seconds (warnings are usually provided by the referee before an indirect free kick is called for violating this rule)
The goalkeeper can drop the ball to use their feet after catching it, but then they cannot pick it up again unless touched by another player (same rules apply as in the pass backs section)
Short Throws
Just like a short pass to a teammate, short throws (or rolling the ball) are helpful for building play through possession from the defense. There are three keys to a good roll out to your teammate:
Release the ball low to the ground to avoid bounces (helps your teammate control the pass).
Roll the ball ahead of your teammate when possible. This means both power to get the ball to them and ahead of where they're facing so that their first touch can continue them in that direction or allow for a quick turn. A ball they have to turn back for is called a "negative ball" because it takes away from building a play.
Step forward with the opposite foot (left foot with the right hand or vice versa) as you are releasing the ball to maintain balance and have space for releasing the ball. This doesn't pose an issue on shorter rolls, but when a goalkeeper has to put more power on their roll balance becomes a bigger factor and they must bring the ball back behind their front leg.
Far Throws
Here are the steps for a long throw:
The starting point for throwing a long throw is how you hold the ball. The hand should be bent to be perpendicular to your wrist with the ball pinched in between and your fingers wrapped around as far as possible.
Next is the movement of your dominant arm (one with the ball). It should reach back and rotate up brushing against your ear. Some goalkeepers find the motion uncomfortable and therefore modify it to a side arm throw, however, this has two issues:
Accuracy is greatly reduced because error in release point will move the ball side to side
Distance is reduced because the direction of force from the torque of your arm isn't in the same direction as the force generated by the rest of your body's movement.
Next is the arm without the ball. As you reach back with the ball, this arm will reach forward in the direction you want to throw. As the dominant arm rotates up the non-dominant arm rotates down to offer balance and generate more power.
Now we have footwork. As your dominant arm goes back, plant your dominant foot and step forward onto your opposite foot as your dominant hand rotates around and releases the ball.
Finally we have the release of the ball. The ball should be released at some point between 45 and 90 degrees past vertical. Releasing higher results in a further throw and higher arc.
Volley Distribution
See the videos below for demonstrations of side volley and drop kick volley techniques.
Side Volley
Punt Volley
Dropping the ball
When a goalkeeper has lots of time, they can drop the ball at their feet and proceed to enter it into play from their feet. As noted earlier in the rules they must keep in mind that once they drop the ball to their feet they cannot pick it back up. This has a couple benefits:
It forces the opposition to challenge the goalkeeper allowing for more time to enter the ball back into play.
Some goalkeepers prefer to distribute the ball from their feet and have better accuracy and/or distance.
As a goalkeeper myself, I prefer to distribute by throwing the ball or kicking it from the ground, because I have significantly better accuracy using these methods and am far less consistent in the distance I get on my volley distribution. Choice of method will definitely depend on your strengths and your team's play style.