The US Constitution does not explicitly authorize privately run militias.
At the time of the writing of the constitution, state's militias were 'well regulated' (by the STATE). "Well regulated' meant that they were called to muster every morning, called to do drills, and try their best to be a military ready body, and these functions were run by the state. They were not 'private militias', per se, though every able bodied male 18 and over, were required to join.
Currently, there are no state laws (I know of, anyway) that specifically authorizes civilian militias to be deputized as official agencies of the state for paramilitary purposes.
In 1787, when the constitution was ratified, state militias were state-run organizations, rather than private militias. The Constitution gives the federal government the power to "provide for calling forth the Militia to execute the Laws of the Union, suppress Insurrections and repel Invasions" (Article I, Section 8, Clause 15), but it also reserves to the states the power to appoint officers and train the militia according to the discipline prescribed by Congress (Article I, Section 8, Clause 16).
At the time of the Constitutional Convention in 1787, the state militias were typically composed of male citizens who were required to serve in the militia for a certain period of time, often on a rotating basis. These citizens were subject to state laws and regulations regarding their militia service, and were organized and commanded by officers appointed by the state government.
While there were some private militias that operated outside of the state militia system, these groups were generally viewed with suspicion by the government and were often suppressed by state and federal authorities. Overall, the state militias were considered to be an important component of the national defense system and were subject to state and federal laws and regulations.
Now then,
The "state militia" as described in the Constitution is now the National Guard as established the Militia Act of 1903, also known as the Dick Act.
Now, you can form a private militia, and call it that, and do whatever it is you want to do on weekends, paint ball, shooting in gun ranges, have a ball, fantasize about being a mercenary, but the above statements are fact.