Private lives can cover everything from romance to hobbies to the type of food your characters most like. These touches add depth to your characters plus if, say, a character likes watching a film, going to a cinema could well form part of the story (even if it is just to show said character having a break from the hurly burly of whatever situation you've put them in).
Private elements of lives can form parts of conversations between characters, give them a reason for meeting at all (and could act as a catalyst for actual story events you do want to include).