In direct combat the private isn't that useful, not in modern day. We have weapons far more devastating then what his scales can protect against that can fire from far away and out maneuver him. In a fight against a fighter jet or any of our big weapons of war he simply goes splat too quickly to be more then a brief tactical surprise. He's pretty strong to infantry, but modern wars are decided less and less by infantry then by big expensive guns that overpower him.
Leushenko answer is great and close to what I would have answered, his ability to hide as a human and thus 'sneak' in pretty massive fire power where it's least expected is powerful. It's less about raw power, more about being able to extend it to places your enemy does not expect and would not allow it. It's also important that he can do it in a way that the enemy doesn't have time to get any of their big guns, like fighter jets, to him before he disappears again.
However, the answer didn't go in to how fully to exploit that power, other then assassination which it mentioned was usually politically less advisable. Actually assassination of semi-important figures may still be a good option. You can't assassinate the head of state, but taking out some of it's high ranking military officers is only going to cause a bit of political difficulty, the tactical advantage of taking out those leaders can be worth it. For that mater taking out lower ranking logistical or leaders at any front right before a major offensive would be useful and generally considered 'fair game', though most of them will be in bases and as said before bases tend to have fighters and other weapons of war near by, making it a risky play since you may not have time to get away safely before they can get their stronger weapons readied to use against you.
However, All the above is just an example of a more general concept, guerilla warfare. Guerilla warfare is probably one of the hardest types of wars to fight, the US had massive technological, numeric, and logistical superiority over Iraq when it invaded, it still left years later with a decent number of lives lost and no real headway because guerillas just couldn't be killed. However, the one limit to guerilla warfare is that they can't get any big guns, their limited to small arms and occasionally small explosives, enough to kill people but not enough to do much major damage to the military weapons.
Private firestorm is different. He is able to do guerilla warfare, as a person he can disappear at any time into a crowed, but he also brings big enough fire power to destroy whatever he chooses to. He can be a threat to larger military hardware, with far less risk then regular guerilla operatives, then disappear back into a crowed.
In any place where infantry is used heavily, which now of days is usually to control areas with civilians like cities, he is very powerful. He can crush military solders and fade away. So long as he makes sure to always pick a place where he can attack and kill fast then get away back into a crowed before the fighter jets and tanks can reach him he is pretty safe. He can single-handedly stop military occupation of any area he is at, and if he picks his targets right could probably strike at lone higher value targets like tanks, jeeps, supply conveys etc, to score larger victories. In fact after awhile he becomes a terror weapon that works just by scaring the soldiers into never going anywhere because at any moment a fire breath instrument of death may show up to crush them.
Of course when he starts this tactic three counter-tactics come up.
- Start looking with the guy with strange eyes. This...won't work as well. There are too many people, and too many easy ways to hide his eyes to make it practical to find him in any decent crowed of people. Plus, if infantry find him he can always go dragon, crush them, then fly away and hide again; so long as there isn't a larger weapon at war beside the infantry men who ask to see his eyes he's still somewhat safe; and do you really see solders being suicidal enough to insist on trying to identify someone that will kill them if they succeed and then likely get away with it?
- Start creating a better system for tracking him when he flies away, using small drones, satellites, and quick responders you can probably manage a system of tracking him in dragon mode and seeing exactly where he 'goes human' and getting both infantry, to deal with screening humans to find the dragon, and larger weapons like fighters to that area quickly. Modern military can use this tactic to make it far more dangerous for him to go dragon anywhere due to the difficulty of loosing his trackers long enough to disappear into the crowed.
- Start arming people with the means to deal with him. Keep fighters and tanks at hand when your doing operations the dragon would likely want to interrupt. More importantly, arm your solders with explosives that can take out a dragon, enough people carry hidden RPGs and your dragon's going to start to be in danger even from 'easy' victims.
However, option 2 and 3 can be fixed easily enough, keep him on the move. Put him in a situation to attack a few people at location A, then hope him on a truck and drive him to a front on the other side of your country to do the same thing there tomorrow. With rapid transportation he can move around a massive area. With such a huge front to attack from it's no longer practical to try to arm everyone with weapons, or surveillance, equipment to deal with him.
Combine with that picking areas of opportunity well and you have a pretty lethal combination. Take out the few guards on a Dam over here and you can threaten an area with liquidy death. Destroy the power lines and generators over another town and leave the base blind and unready to deal with your infantry front. He can be pretty disruptive, though it takes a pretty extensive logistics to figure out where he should be traveling, and some of this can be done by smaller elite cells almost as well as he would do.
But there is one step better, move him away from the front lines entirely, send him on a counter offensive. Have him travel to your enemy countries home; surely you can figure out some way to sneak one operative past their boarders; or just have him bust his way in through a border somewhere and then hide.
Once inside the territory he is actually quite safe. Past their front lines they won't have the big weapons of war, the things capable of killing him, so there aren't many things that are a major threat to him; other then being spotted and tracked so well then he can't lose (or destroy) his trackers in the time it takes to fly a fighter from a base a ways away from him (and even then, the enemy is going to be apprehensive firing missiles at a dragon flying right above a crowded city of their own civilians...
Now if your evil you can start a terrorism campaign that would be very effective, but at that point assassination is an option to you and well...your be pretty effective until every nation declares war on you. (though...if you arrange to say he is awol terrorist and not supported by your nation perhaps you can get away with some of it).
However, even if your not evil you still have targets to attack. The military already agrees that there are strategic targets that are fair game for attacks even in civilian areas. Any place that manufactures weapons of war will be a valid target, for example. Power plants I believe can be, training grounds if you find any that aren't too defended by bases with fighters, etc. He can simply drive around the nation, attacking completely random strategic infrastructure locations at seemingly random. Make sure to never stay in one place for any length of time, and to always have a plan for how to 'go human' and get away quickly before helicopters can be set up to start following you (if you are followed by helicopters he can destroy them, but since they would be civilian police it's getting to be a little more diplomatically grey; best to take them out without killing fliers when possible).
The only thing he really needs is sufficient source of raw money to be able to buy bus tickets to get around. He needs to travel long distances between targets to keep everyone guessing, but that isn't to hard to do in a modern country.
He still has the danger of being tracked long enough that the police can get a jet to him. The biggest danger here is if he is tracked without realizing it, smaller drones he doesn't realize are following him. If he can't change his human appearance pictures of him will circulate after awhile, but he can hide his face and still get around pretty well, the country is just so huge that there are too many people he can disappear behind.
If he manages to keep this up long enough without being caught eventually the terror aspect will result in people afraid to work on building weapons of war because he may show up and destroy their building. That affect can do far more harm then his actual attacks do, it's all psychological. The important thing would be to make sure he attacked only strategically 'far' targets to keep the moral high ground of "were not terrorists, were just attacking valid targets with the weapon we have while trying to minimize civilian loss of life". He should try to minimize lost of life, important diplomatically, but he still works as massive psychological warfare even if he is doing that.
I can't say how many infrastructure buildings he can get to before someone manages to track him, but frankly it won't take many before he starts to really cripple the military infrastructure and force them to bring a disproportionately big part of their military back simply to assure their civilians that they have people ready to deal with the big scary threat.