In Dredd we occasionally see privacy, and more specifically information privacy, appear as a topic within the movie. One example of this is when a civilian refused to give the two judges access to a certain room. It's certainly possible that like police in the United States, the Judges cannot legally enter a private residence without probable cause. Though it should be noted that this may have been partly due to the judges' inability to force themselves into the room, especially due to Dredd's apparent lack of respect or care toward civilians.
Additionally it is important to recognize that Mega City One is extremely densely populated. The residential complex that the majority of the movie takes place in, Peach Trees, is composed of hundreds of small apartments. These apartments are locked and appear to demonstrate some degree of personal privacy. However, with the entirety of the mega-city being composed of massive apartment-style buildings, it is doubtful that most civilians would be able to own private property.
Furthermore, many times throughout the movie we see how the judges are able to track citizens' location at all times using security cameras and drones. Facial recognition is also used to aid the judges. This is a high level of surveillance, and subsequently a very low level of privacy. Although we do not directly see it in the movie, we could also assume that this lack of privacy may also extend to citizens' online activity.