This is a good point folks often forget. Those feathers aren't as water resistant as say a duck's, but they do protect the skin a lot. Most birds can look pretty wet, but if you wiggle your finger down next to the skin, it's nice and dry.
This actually goes for a lot of animals--my dogs can look drenched, but their skin is dry. Horses, cats, etc are the same.
Depends on the breed though. Some dog breeds have very thin coats with NO undercoat (i.e. Dobermans for example). They are extremely cold sensitive and they HATE getting rained on, without an undercoat the cold/wet goes right to their skin.
I noticed over the years the thicker coated dogs often don't mind the rain but the thin coated dogs will many times avoid it at all costs! Same concerns may apply to some chicken breeds that have been altered dramatically by selective breeding.