Goblin Sharpshooter does not have summoning sickness.
Steps
Activate Goblin Sharpshooter by tapping it, dealing 1 damage to a creature an opponent controls.
The creature dealt damage dies as a state based action due to having 0 or less toughness, triggering Goblin Sharpshooter, untapping Goblin Sharpshooter.
Repeat as desired.
You may also repeat this loop whenever a creature enters the battlefield under an opponent's control.
Results
Destroy all creatures that enter the battlefield under an opponent's control.