In the early days (1861) some units were equipped by a wealthy patron, and some officers and men may have brought their own firearms into service. One way or another, it made financial sense for gun owners to be paid for personal weapons in government service, so these soon found their way into CS ordnance inventories. Those who hung on to their own weapons would have had to procure and pay for their own ammunition.
Inspection reports for 1864 will state the number of PFD and arms on hand. The inspector then notes the number of arms required. For instance, if PFD is 250 and number of arms on hand is 200, the inspector will usually note 50 arms required. Maybe I'm wrong, but the math usually leaves no room for privatly-owned arms.
The inspector also notes condition of arms on hand as good, fair or poor.