No. We know some things they do. We don't know what else and how much they do.
Now most of the general public knows the nature of their work and some of the details surrounding it.
Ideas surrounding freedom, liberty, and privacy are very complex, easily confused, and often forgotten until its too late. The laws around these things are mechanisms which help protect what you're country supposedly holds dare.
But in answer to your actual question, would you believe them if they said they hadn't?
They might undergo some reform, but the government apparatus has been far too reliant on many aspects of their work to actually shut them down.