Do Chinese water dragons and bearded dragons eat the same food?

If you are refering to the fruits/ leafy greens that are safe to feed iguanas, then yes. I wouldn't give them any of the premade Iguana food that you can buy at the pet stores though.

Chinese Water Dragons can eat the same fruits and veggies, but usually don't like them too much and will only pick at them rarely.

Make sure you are feeding mostly insects/whole prey: crickets, mealworms, waxworms, phoenix worms, butterworms, silkworms, clean feeder fish. Some people give pinky mice, but only when they are around adult size. Good luck. :)