The way we do it in my setting is that the group officially works for the government, but in an undercover fashion. Their mandate states that they must undertake certain assignments for the government of an investigative nature, but if they happen to discover something on their own they are free to investigate or explore it as long as they do not conceal their findings from the government (the Byzantine empire, and sometimes they are jointly employed by both the Byzantine Emperor and/or military, and the Orthodox Church) or their employers.
In their free time they can work for whomever they wish as long as they keep the government apprised of their activities, and those activities do not counter the interests of the government (such as, they cannot be employed by the Persian or Bulgarian kingdoms, unless they are covertly working as espionage agents against those kingdoms - as double agents).
They are all professional and trained at government expense, and paid at government expense for government work, and they have earned things like tax free estates, and titles, and they are often equipped at government expense, and can keep a percentage of all goods recovered from any mission/adventure they are sent on by their employers, in addition to their pay. What they recover is considered their hazard bonus. They can keep all they recover from outside employers, or freelance contract work, as long as they catalog and report it to the emperor and/or Church.
However their actual work for the government is secret and they are not publicly connected to it. Rather they are granted minor nobility titles and other privileges as well as cover stories as a public identity.
This arrangement allows them to work together (on an ongoing basis), to have a shared backstory, even though they are originally of very different backgrounds, and yet still allows them to work freelance or as independent agents when not on official assignment. It also allows them to maintain their secrecy and escape scrutiny from much of the public. They are often employed 6 to 9 months out of the year for the Byzantines and/or Church. Depends on what is going on. Works very well for my party, although there are also other parties operating differently. Promotes group and unit cohesion and morale, but alos allows them to be able to work independently when necessary or they so desire.
But if they are on assignment or are operating outside the Byzantine empire then they are effectively on their own.
Don't know if that helps ya or gives ya any ideas, but it's what we do.