This outcome was inevitable.
All the talk about flags is one thing, but when they say "dungeon on the other side of the mountain", what they really mean is "the entrance is on the other side of the mountain" and more particularly "you're both digging inside the same mountain".
I completed both Dungeon Keeper games back in the day, and funnily enough they play out VERY much like this story in certain regards. One of them being that usually you will encounter open caverns in the process of trying to expand your own dungeon, and quite often those open caverns connect directly to other dungeons or the hero portal.