There's a law that prohibits all video monitoring of public spaces, and a register where you must register your video cameras if you're a business owner. Video surveillance in Denmark has a maximum legal retention of 28 days, unless there's an ongoing investigation.

Considering that dash cams mostly monitor "public spaces" and are moving around, the legality of them have been questioned multiple times. They are however also becoming more and more common, so I'm guessing they will eventually be allowed with a relatively low retention, like 1-2 days, enough to get footage off of them in case of a crash.