Game of Thrones answered this question in the first five minutes of the first episode, and almost everyone forgot.
The series opens with three rangers beyond the Wall. The White Walkers butcher two of them and deliberately let the third go. Will rides south, tells everyone what he saw, and gets beheaded by Ned Stark for desertion. That is the show's opening act.
A season and a half later, they do the exact same thing to Sam. A Walker on horseback looks directly at him, screeches, and moves the army along.
This is a real military doctrine, older than the books. Genghis Khan's armies deliberately released survivors from cities that resisted. The refugees ran ahead spreading stories of what they saw, and the next city surrendered without a siege. Fear traveled faster than the horde and cost nothing to feed.
For the Walkers the economics are even better. Every man who flees south eventually dies somewhere, and every death is a future recruit. Killing Sam gains one corpse now. Sparing him spreads panic through the entire Night's Watch, and they collect the body later regardless.
An army that raises the dead loses nothing by letting you run. You're inventory either way.
The darkest part is what the living did with their messenger. The first man the Walkers ever spared to deliver a warning was executed for delivering it, by the most honorable man in Westeros.
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During the British Raj, officials grew concerned about the rising number of venomous cobras in Delhi. To tackle the issue, they introduced a bounty for every dead snake.
At first, the policy seemed successful, many cobras were killed.