Which is just a complicated way of saying you kill them because they're in the way.
I'm currently thinking about a game wherein you are (primarily) monster hunters, rather than slayers. This entirely changes how the players relate to the monsters - they are the driving purpose, they are the main event. Consequently, how we as DMs work with the monsters has to change to reflect this, thematically, in presentation and mechanically.
(It'll also change the XP calculations - though I'd still probably link it to "treasure", but the treasure being the reward money for claiming the bounty)
Thematics
There's a couple of schools of thought to be applied here - what does the monster mean, or represent*. Integrating monsters into the (super)natural fauna of the land gives them context, a place, relationships with the rest of the system but simultaneously runs the risk of making them a bear or wolf with magic powers, which is a bit shit. This links heavily into presentation, discussed in that section. However, you could argue that animals are not monsters, that animals are an expected element, a known quantity. Monsters are (in my view) essentially aberrations of the natural order.
Taking on this concept of the monster as an aberration, we can play with some pretty fun ideas - they represent a wrongness in the area, whether that be social or environmental - the spirit of your murdered brother, or woodland furies emerging to slaughter woodsmen. With this, you introduce the concept of having two immediate routes to solve the issue - hunt and kill the monster (treat the symptom) or attempting to solve the issue causing the aberration in the first place (cure what ails). This concept is used in The Witcher 3, which is a really good game you should try out - although you do sometimes end up fighting the beastie after solving the problem, albeit in some modified way, such as a weakened state, with greater foreknowledge, or with some form of specialized preparation informed by the root cause - put the spirit to rest with their original murder weapon, or prepare an oil with the weeds atop their unhallowed resting place.
Presentation
The monsters themselves must be presented according to their themes - the must fall into place with their respective systems, whether this is naturalistic "monsters" having nesting and food, or the aberrant monsters presented in such a manner to reflect their nature, and the ill they represent. Either approach can give the campaign a more "folky" feel, relying on the local population, their legends and rumours, a lot of spent in the rural areas rather than the cities and towns.
Divorced from this, the monsters must be capable of being hunted. This can mean innate stealth, necessitating tracking, great speed or flight, requiring some manner of trapping them or attacking them at rest, or just being them so tough and terrifying that a cunning plan is needed to take them down.
Mechanics
The above points should generally somehow be reflected mechanically - with a lot of these, as with most OSR & OSR adjacent stuff, is in rulings and presentation rather than encoded into rules specifically.
A principal challenge with a hunted monster is ensuring the actual confrontation is interesting - I'm sure we've all experienced the players blitzing a single powerful opponent to death in a single round. HP Bloat is not enough to give this feeling of a genuinely dangerous monster worthy of hunting - elements such as speed and stealth, as with presentation, are key. Tapping into the folky nature of the game, having very specific weakness (which, of course, can be discovered by characters investigating sufficiently) are a both a classic element of such legends, as well significantly toughening up the monster itself.
Of course, an infestation of monsters, rather than a single, powerful creature, entirely side-steps this issue, but changes the entire relationship of the hunt - an extermination rather than a hunt. However, many of the other points above still apply, such as infestation-as-aberration, especially in the case that the infesting creature is otherwise normal in many respects.
*Not that I generally ascribe meaning to anything in my games in terms of real-world stuff, but more a sense of history or integration, or going the other way, a negation or lack of history/integration, and indeed, meaning.
