I don't think people are that naive. The reason for that arrangement is simply that it is only necessary for the NSA to collaborate with one Danish intelligence

I don't think people are that naive. The reason for that arrangement is simply that it is only necessary for the NSA to collaborate with one Danish intelligence agency (foreign intelligence, which is probably forbidden from spying on Danes), rather than two (foreign and domestic). In a larger sense, the arrangement might, to a degree, defeat the purpose of having separate foreign and domestic intelligence agencies, but that is a more abstract notion. I guess we'll see what the Danish public will think or do about.