- If a soldier reported the lost during combat and it has been really lost and another soldier (or private) found it and brought it back. What is the legal status of the material?
A private in the Army is a soldier.
All unit property the government owns is accounted for by a hand receipt holder. Responsibility for maintaining accountability of property is further delegated down to an armorer, who absolves himself of responsibility by using an arms room card (temporary hand receipt) to issue out the ACOGs to the soldiers who are primarily responsible for them. It is possible that someone can lose one in working order and have it declared a field/combat loss, though entirely unlikely that they wouldn't also be facing at the minimum a bill for the cost of the ACOG minus a percentage of depreciation, and at the most UCMJ action for failure to maintain their equipment.
It is also possible, though also unlikely, that somewhere along the line in the logistics chain somebody pilfered one and declared it a loss. Again, unless you're talking about a commander who just doesn't care about supply discipline and who likes paying for equipment he didn't lose, this probably isn't too common.
The bottom line, and in answer to your question, is that any property the government purchased belongs to the government until the government decides that it doesn't. Even if someone lost it and paid the cost minus depreciation back to the government, THE GOVERNMENT STILL OWNS THE PROPERTY.
If I lost an ACOG and paid the government back for it, and then later found the item and returned it to the government, I might be able to get my money back. But there is no scenario that ends up with me legally owning the item.