My house is the one with the red door on the right.
My house is on the right, with the red door.
Are both sentences idiomatic and do they have the same meaning?
Both are fine. There are potentially a couple of different meanings of the first sentence. It could mean "my house has a red door and my house is on the right" or it could mean "my house has a red door and the door is on the right".
In context there would be no ambiguity. If you were saying this you would be looking at your house, or a picture of your house, and everything would be clear. In context, both are acceptable sentences.