Notoriety is gained by public activity of an unusual, but mundane, nature. Having a shoot out in town, carrying around (or using) military weapons, driving around in an undisguised military vehicle (or a civilian vehicle with military weapons), being seen pushing around civilian authorities or breaking laws, etc. Being seen harming civilians (or things that look like civilians…) or causing collateral damage is a huge boost.
Having a high notoriety makes life difficult for the monster hunters, but in a mundane way (initially, at least). The people who sell the monster hunters their military weapons and other gear are not supposed to make it publicly known that they do what they do, so they will slow down or stop. Onlookers, busybodies, and reporters will appear faster, when greater regularity, when the monster hunters have a high notoriety, causing all kinds of problems. High notoriety makes it easier for the monster hunters to be tracked down by anybody who wishes (whether for good or for bad). If monster hunter notoriety gets high enough, the MCB will cut all ties, and the hunters could find themselves in serious legal trouble.
Notoriety is lost by staying inconspicuous. It can also be reduced by removing/hiding evidence, or even leaving no witnesses, as well as being mitigated by clever plans or communication with witnesses.