In the modern parlance paid civilains carrying weapons and fighting in war zones are “security contractors” or “private military contractors” (PMCs). They were previously referred to as mercenaries or “soldiers of fortune” but those terms have a sour connotation.
It depends who they are working for. In Iraq and Afghanistan, PMCs are employed directly by the Department of Defense, by the CIA as paramilitary operators working with the Special Activities Division, or as private security to other construction, intelligence, or logistics contractors. PMCs are not supposed to engage in offensive operations per US and international law (specifically Military Commissions Act of 2006 and the Geneva Conventions) but they often do, especially in the case of special operations that lack official sanction. PMCs are not part of military command structure but may be integrated for certain purposes such as intelligence or logistical support and given a simulated rank (for the purpose of heirarchy, although no civilian below the Secretary of Defense can give orders to military personnel), and could be subject to arrest as unlawful combatants if they are performing illegal operations in an area of operation that the military controls. Generally speaking, PMCs have few legal protections and often operate in a legal grey zone.
Armed private civilians not operating under contract are by definition illegal or unlawful combatants and may be subject to laws in both the country they are operating in as well as their native company. You can’t just waltz into a war zone and start shooting people or looting because that would be murder and rapine. You have to get Uncle Sam’s stamp of approval first.