Every dragon in my campaign is a powerful entity. I give them all very high Intelligence scores, so even the White dragons are smarter than most of the humans who bother them.
I generally leave them backstage, though there are quite a few of them. All of them have the ability to take a variety of humanoid forms, and they like to use this ability to take positions of power in humanoid realms. They are cunning, political, and typically supported by many, many supporters, followers, retainers, and minions. Most of the time, the PCs aren't even aware they're dealing with a dragon, unless they have some way of seeing through their humanoid forms.
Half-Dragons are quite a bit more common, and can be encountered nearly anywhere. Unless their draconic parent is openly a dragon, they are typically unacknowledged; they adventure to earn power and wealth, either to impress their parent or to surpass them.