I remember William explaining about how weird he found it when they came to see the flowers at KP after Diana's death, with people pawing at them and crying for this woman they didn't know, as if they had a right to share his very private grief.
It must have been very difficult for him to deal with. However, feeling an emotional connection to someone you only know through the media is the very foundation of the monarchy. If that connection fails, the monarchy's days are numbered.
It could be worse. Early modern royals were expected to eat, get up and go to bed in public. The balustrades in front of the beds at Versailles were intended as a form of crowd control, since as many as 60 people might show up to watch Marie Antoinette get dressed every morning. There were dozens of people in her bedroom the first time she gave birth, and it was so crowded that she passed out for lack of oxygen. She would consider the life of our current RF a haven of privacy by comparison.
Kate and the paps: yes, she was pursued relentlessly before her marriage, and not only in the street, they chased her in her car. She once pulled over and asked them to stop as it was dangerous and they said if she didn't like it she should stop dating William. Sorry no link as I read it in some royal bio, can't find which but if I do, I'll post the reference.
In other news, I'm pretty sure I caught a glimpse of Princess Anne today! Being driven up Tottenham Court Road. Unfortunately I can't find anything in the Court Circular to confirm where she was today.