In Liam O'Flaherty's "The Sniper," the main character always gets the attention. This question is awesome. Of course, we don't know a great deal, but what we do know comes from the "main" sniper's interaction with his brother. Let's refer to the sniper as our main character, and the brother as our second character.
First, we know that the brother is observant: he immediately sees the quick lighting of the sniper's match. He is an excellent marksman: he responds immediately with a bullet that forces the sniper to put out his cigarette and crawl away. When the sniper peeks over the short wall covering him, the brother sees him and swiftly fires again. The brother is good at his "job"—he finds cover to protect himself and watches his enemy's every move. When the sniper kills the man in the tank and the informant (the woman), the brother aims and hits the sniper.
It is at this part of the story that we realize that the other sniper is probably not quite as experienced as the main character. When the sniper puts his cap on the barrel of his gun and raises it, the brother shoots and the sniper drops his weapon over the side of the building, and then lets his arm droop as if he has been hit. Over the next few seconds, the sniper allows his arm to slip out of view as if he has died.
Believing he has taken out his enemy, the brother stands and it is in that moment that the sniper, with his revolver and uninjured arm, fires at the other man. The brother is hit and falls, hitting the ground below. We know that the sniper first "utters a cry of joy" and from this, we can infer that it is because he has accomplished a difficult task—killing a worthy adversary and enemy. (He does not react this way when he kills the man in the tank or the woman.) However, we read:
The lust of battle died in him.
From this we can infer that the challenge of facing another excellent marksman was thrilling, but in the other man's death, there is also the loss of a valiant opponent—a man to be admired—even if he is on the opposite side. The sniper's reaction provides us with the knowledge that he held the other man in high esteem and is sorry for his death—perhaps acknowledging the passing of a "fellow warrior."