In the sense that the two planets move together and has intelligent beings with their own forces and particles, but they just do not interact with (are invisible to) us?
In principle you could, and there are some theorists that believe that dark matter has its own interactions. However, unless there is some fundamentally wrong about our understanding of earth's interior, we could not have a dark matter planet superimposed with the earth, or we would feel the additional gravity force, which nothing indicates it is there.
No. Every piece of evidence that points to dark matter existing simultaneously points to it interacting extraordinarily weakly with everything else, including itself. A dark matter particle in all likelihood could pass back and forth through the Earth a billion times and still emerge untouched by all the matter it passed through, whether that matter were itself "dark" or not. Thus you would never have small-scale structure with dark matter. It won't form boulders or specks of dirt or even molecules, so there is no chance of building up a planet of the stuff, much less of supporting life.
If you want more direct evidence for this point of view, the earliest indications of dark matter came from noticing that stars on the distant outskirts of galaxies were moving much faster than expected, indicating that there was more gravitational mass inside their orbits. The best explanation is a very diffuse halo of "dark matter" overlaid on the galaxy, but very much larger than it. If dark matter could interact with itself in any significant way, even if it could not interact with normal matter at all, it would be able to dissipate heat and transfer angular momentum, causing this halo to collapse to the size of the galaxy, just as the normal matter did. It can only have the diffuse profile we observe if it doesn't interact with much of anything in any significant way (other than via gravity, of course).