Most shotgun powders burn fast, that is why you can use them in some pistol cartridges.
Shotguns work on low pressure, and have very large bores for their relatively small powder charge, this results in a lot of surface area to burn when hit by the primer sparks, without a very high maximum PSI to reach.
A lot of the powder is burned in the first 6-8 inches with many powders, and by 12 inches you would see the majority of the velocity.
This of course is based on the burn rate of the powder, slower may benefit from a longer barrel.
In reloading you will find that heavy rounds for a given caliber lose the least "energy" from a reduced barrel, because they can build up to the peak pressure in less room. While high velocity light rounds benefit greatly from added barrel and lose a much larger percentage of energy as you remove barrel.
Shotguns put out huge payload at relatively low velocities, and so likewise need minimal barrel to reach or maintain peak pressure.
Old black powder shotguns needed more barrel because black powder burns slower.
A quick search and I noticed a lot of data for over 18" but not as much for under, obviously because fewer people have NFA firearms.
One of the top google finds was this guy using a chronograph, his post is filled with a lot of profanity, which unfortunately detracts from the otherwise useful information:
http://forum.saiga-12.com/index.php?/topic/19103-barrel-length-vs-velocity/
Without the added profanity here is the chronographed data:
birdshot- 12 gauge, 2 3/4 inches, 1 oz shot
12" barrel average FPS- 1040
14" barrel average FPS- 1069
18" barrel average FPS- 1114
19.5" barrel average FPS- 1114
20" barrel average FPS- 1108
28" barrel average FPS- 1159
Then using Remington 00 Buck-
12" barrel average FPS- 1121
14" barrel average FPS- 1127
18" barrel average FPS- 1216
19.5" barrel average FPS- 1205
20" barrel average FPS- 1182
28" barrel average FPS- 1250
This variation coincides with another graph I located on THR posted when someone asked a similar question:
(posted in case the image disappears) 7" is about 1150FPS, 8" it is about 1175FPS, and 9" it is 1200FPS, by 12" it is about 1350 FPS, by 18: it is over 1400FPS, and from about 20-30" (with 20" clearly an anomaly or severely backbored barrel or with a choke issue going on) it stays around 1500 +- 100FPS and some margin of error.
So for as handy and portable as the short barrels are they give up little with typical powders. The main benefit in hunting uses is a certain length swings better for people and makes hitting small moving targets at a distance much easier.
The long barrels do dramatically decrease the decibel rating as well, a few inches makes a huge difference in the sound category, not just slight but huge.
The perceived muzzle blast is also greater the closer the bore is to your face. The shorter the barrel the closer you are to the bad side.