Handgun Throw is a secondary skill that lets you throw single-hand weapons for damage at the cost of 1 star in the Shootout game mode.
By pressing the V button, a charge meter will appear and quickly build up, throwing your weapons as soon as it builds up completely.
You don’t need to wait until the meter is fully charged to throw your weapon, simply pressing the primary fire button will throw the weapon at the current charge (assuming you’re wielding a single weapon).
When dual-wielding, pressing primary or secondary fire while it is still charging will only throw your left or right-hand weapon, respectively. Letting it charge completely will throw both your weapons.

Max damage seems to be reached at around 50% charge, charging it further will only make it travel faster
Distance doesn’t seem to affect damage.
A thrown weapon can ricochet/bounce off a wall or the ground but will still deal damage upon hitting a target.
Different weapons deal a different amount of damage, which also depends on how many bullets the weapon is loading as it is thrown:


The Ghost Gun (knockback-only, no damage weapon from Halloween maps, looks like the Sawn-off but with green wooden parts) has the same damage and notoriety values as the Sawn-off

*The Sawn-off/Ghost Gun/Mare's Leg/Peacemaker seem to not apply the same notoriety every time, randomly switching between the 2 values

The Portable Whiskey also can be thrown for damage, although i couldn’t get the exact damage values for the Whiskey Jug as it refills over time instead of having to “reload” but i did get the minimum and maximum damage values for it:


According to Update Notes from March 12 2018, getting hit by a thrown Whiskey Jug will add some drunkenness to you, but I haven’t confirmed that yet as all my testing was done with frozen bots

I guess the only things i have left to say is to clarify some of the previous things i've said:

Originally posted by Foreign Creature:

lets you throw single-hand weapons

By that, i mean only the weapons you can use to dual-wield can be thrown for damage.
Here's the list of weapons that CANNOT be thrown for damage:

• Bow
  
• Smith Carbine
  
• Yellow Boy
  
• Coach Gun
  
• Spencer Carbine
  
• Pump Shotgun
  
• Sharps Rifle

The following cannot be dual-wielded but can be thrown to deal damage even WITHOUT the Handgun Throw skill:

• Knife
  
• Hatchet
  
• Dynamite
  
• Black Dynamite
  
• Machete
  
• Dynamite Belt
  
  As i've said in the very description of this guide, i will not go into detail regarding the damage of these weapons.
  
  Thanks to Steam user Yewie and Reddit user u/MrJAG_Fistful i have noticed not all weapons give the same notoriety and have updated the guide.