A level of radiation significant enough to draw official attention would also be significant enough that the people in that building would not want to be accidentally exposed to it.

Certainly if the feds came to my house, pointed out that my radiation levels were high and then discovered that (like about 1-3% of US houses) I had a high enough background level of radioactive materials to constitute a significant risk of cancer, I'd personally appreciate it.

I'm with Hugh. Checking the radiation level of my house isn't going to compromise any personal information about me. I object to it less than, say, taking a picture of my house using a satellite. (Which pictures are commonly used by municipal governments to spot people who have been doing unauthorized construction and aren't paying taxes on their new amenities.)

Cheers,
Ben