There's a difference between privacy and anonymity. Privacy keeps people from seeing the stuff that you don't want others to see. It has little to do with freedom of speech. Anonymity is putting stuff out in the public without them knowing who said it. Yes, it's freedom of speech, but it's also freedom from accountability.
In a fully functioning democracy, you should be free to say whatever you want openly without being threatened. If you can't make an argument openly, I would immediately question why: is it because society is unjust, or because you can't stand behind the argument?
I would argue that you would still need privacy, in so far as to allow an amount of time in which you could form your dissenting thoughts in private, write them down, iterate them before you go to the street corner and announce them to the world.
If everything was in the public space or monitored then people with dissenting ideas may not have the will or motivation to iterate on them as is necessary to create a well reasoned thesis, and hence they would fall flat or be cast out as just another nut job.
Oh, no - I'm not trying to downplay the need for privacy. I'm saying that privacy and anonymity are two separate things and people too often mistake one for the other. Privacy is everyone's right, but it has nothing to do with dissent. Blasting your opinions out into public while shielding yourself with a pseudonym isn't privacy.
I consider that anonymity and pseudonymity are aspects of privacy. What's private there are the associations between activity and identity. We need "privacy for the weak and transparency for the powerful" (see Cypherpunks: Freedom and the Future of the Internet by Julian Assange et alia).
In a fully functioning democracy, you should be free to say whatever you want openly without being threatened. If you can't make an argument openly, I would immediately question why: is it because society is unjust, or because you can't stand behind the argument?