I wonder who would win

I think a dragon would have a pretty good chance of beating a WWI fighter plane. If it's a trained dragon with a rider, an exponentially better chance.

The dragon has more ways of attacking the plane- it can breathe fire on it of course, but it can also dive on it and slash it up with its claws. Since WWI fighters were mostly wood and canvas, a dragon striking like a falcon at a single point on the plane would be devastating.

The fighter pilot, of course, has his guns. I'm assuming a dragon's scales are at least as tough as an elephant's hide, and the .303 Vickers machine gun used in the Sopwith Camel and SE5 is adequate to kill elephants...

The 7×57mm Mauser used in the Fokker Dr. I's gun was also able to kill elephants. WDM Bell used both to kill over a thousand elephants. HOWEVER, he recommended hitting certain areas on the elephant; hitting other parts of the body may kill one, but it would take a lot longer.

He specifically recommended the brain, the heart & lungs, and where the windpipe enters the body for the quickest kill, although there are other spots to hit if you don't care about killing humanely (shoulder blade will break the leg, or wing in the case of a dragon I suppose)

These shots are hard enough for a hunter on the ground with the elephant romping around slowly. In air combat, it would be EXCRUCIATINGLY difficult to place a shot in such a manner. If a dragon's scales are substantially stronger than elephant hide, they may even deflect bullets.

Still, I can't imagine any dragon resembling a real animal would be immune to bullets, and a skilled biplane pilot could kill a dragon if he hits the dragon in the right spots. A dragon is a big target; he'd hit the heart or lungs eventually, or shred its wings.

Shredding the wings might be the most effective way to knock a dragon out. They'd be barn-sized targets (maybe literally) and hitting them would hurt. Bats have nerves in their wings that allow them to detect airflows; I'd imagine dragons would also be able to "feel" their wings.

One or two punctures in the wing wouldn't affect its ability to fly, but stitching it with gunfire would, especially if the stream of bullets shatter the fragile hand/finger bones. Dragon bones would probably be very light and honeycombed to aide in reducing flight weight.

There wouldn't be a good way to armor the wings either, assuming we have a rider on the dragon. The body and neck could theoretically be armoured the same way horses are armored for cavalry/police work. Greater emphasis on belly armor, as that's a big target in the air.

If the dragon is intelligent like Draco or Smaug, he's his own pilot and correspondingly would have human-like qualities of bravery and wit. A more "realistic" dragon, like the Game of Thrones dragons, would be animals. They might bolt at gunfire and would need heavy training.

Still, if the dragon is either intelligent in his own right or trained with a rider not to bolt at gunfire, I think a WWI fighter plane is neck-and-neck with a dragon. I think it could go either way, depending on, as all dogfights do, the skill of the pilots and a bit of luck.