Basic Information
Bullet monkey is a slang term for a low skilled (semi)professional gunman - the sort of person than ends up serving as a mook or redshirt in many settings.
Exact skill levels will depend on plot, setting and system, but training at the stormtrooper marksmanship academy is to be expected. It is entirely reasonable for the NPC in question to be mechanically more dangerous without their firearm - and equally credible for them to use it anyway. This can be a case of truth in (television) as absent a degree of skill at arms training which very few people receive and maintain - even in a well armed society like the USA - a lot of people will spend most of a firefight missing their target (actually even people with quite a lot of training can spend quite a lot of time missing), especially when armed with naturally inaccurate weapons such as pistols. Many people who are reasonably proficient in melee combat for one reason or another will, however, psychologically bond to a firearm and try to use it in preference to melee, even if they cannot hit a barn from the inside whilst using it1. Thus, given the right circumstances, most private security personnel, the majority of armed criminals and a good chunk of police officers (assuming a routinely armed force, and as in the UK where the only armed officers are SWAT equivalents) will qualify for bullet monkey status. Many soldiers may even fit into the category in context.
Training in context may also include tactical skills as well as skill at arms - a mediocre shot with a rifle who keeps low, uses cover and takes time to aim can make a terrible mess of a group of opponents who stand around firing wildly in his general direction. The bullet monkey is prone to full automatic fire whenever he has the option, and may otherwise stand like he is on the firing range or strike a gangsta pose before blazing away.
The bullet monkey is distinguished from the true amateur by his (semi)professional status - both may be equally incompetent, but the average Joe or rush hour vigilante lacks the employment status to be a good example of the breed.