Quite the opposite, free speech and privacy demand each other and cannot exist without each other.
Consider these two scenarios, taken to the extreme to elaborate a point.
Unfettered free speech, with zero privacy.
Total privacy, no free speech.
Both examples are impossible in a human society (maybe in a robot society).
In scenario one, there can be no free speech if you have no right to privacy. The concept of chilling comes into action. If humans were robots, one might say that zero privacy would encourage and illuminate perfect speech because there would be greater total information and understanding. However that is not how people function. When faced with zero privacy, people will always self-censor, often dramatically. We can see this in practice time and time again, in numerous countries throughout just the last century.
In scenario two, if you have a government enforcing zero free speech, there's no chance they're going to simultaneously grant their people complete privacy (they can't, they have to constantly monitor speech).
The point of these examples, is to show that one can't exist without the other, not in actuality. As one erodes, the other will go with it.
A government willing to use extreme violence (which is what it would take) to stop all privacy, is not going to give you free speech. No such contradiction can exist for long. Any government willing to use violence to stop your freedom of speech, is not going to respect your privacy because they'll need to monitor you constantly to check against your speech (not to mention a violent government in reality simply isn't going to respect individual rights anyway).
