As usual, I can only speak for the Roman period.

In the imperial era, hundreds of men were employed as professional or semi-professional beast hunters. Their primary function was to capture animals alive for the arena, but they were also called to drive off or destroy animals that threatened crops or farmers.

Many of these hunters were Roman soldiers. Detachments hunted lions in Algeria, bison in Bulgaria, and everything between. A few soldiers, in fact, became semi-professionals, and received titles like ursarius – bear hunter. An inscription informs us that one of these men, a centurion on the German frontier, managed to capture 50 bears in six months.

However, at least in some provinces (above all in North Africa), professional hunters, often organized into guilds, did the majority of the hunting. The hunting guilds could be contracted by everyone from the members of a city council to the emperor himself to supply a given number of animals by a set date. At times of particularly high demand, guild members might be aided by freelance hunters, who probably collected a bounty for every animal they captured.

The driving motivation behind most of this hunting, as mentioned, was to collect animals for the venationes (beast hunts) that dominated the morning events of many games. But the same men who gathered animals for the arena could easily turn their talents to destroying pests or predators.

Some of the animals they had to cope with, admittedly, were a little less intimidating than the monsters in the Witcher series. In fourth-century Egypt, for example, a priest requested nets (and apparently some skilled hunters) from the commander of a local cavalry detachment to deal with a large herd of gazelles that had been devouring the crops of local farmers.

Farmers on marginal lands in North Africa, whose livestock was regularly attacked by lions, seem to have regularly called upon professional hunters and/or Roman soldiers for help. Such expeditions were routine enough to be described by the third-century author Julius Africanus, who outlines how professional trackers (vestigatores) from the legions would hunt lions to their lairs (Cesti 7.14).

The services of the soldiers and professional hunters in clearing the countryside of dangerous animals were widely recognized, most notably by the late antique poet Luxorius: "The countryside marvels at the triumphs of the amphitheater, and the forest notices that strange wild beasts are there [in the arena]...the fertile land loses nothing, and the plants grow in the greater abundance, while all the wild beasts fear their fates [in the arena]" (60 [Rosenblum]).

On Roman methods of hunting wild animals, I refer you to my page on the topic.