I don't believe that privacy is necessarily important per se. Rather, I believe that it's a foundational violated right.
Rights are (not coincidentally IMO) almost always conceived and expressed backwards. That puts the onus on those who would defend a claimed right. I think the onus rather obviously should fall on those who would violate a right - they are the ones who are acting in a specific way, so they are the ones who need to justify their actions.
If an individual lives in complete isolation, they have a complete and total and unchallenged "right" to privacy - it's literally impossible for anyone else to breach their privacy. It's only with the addition of other people that the matter becomes relevant, and only with an attempt by another to breach their privacy that it becomes a point of contention.
So again, and really rather obviously, that other has to be able to justify their breach of privacy, since it's specifically them and their actions that have made the issue relevant.
And at that point, it's really a very simple question - who has a greater right to control over the details of an individual's life - that individual or some third party?
So it's not so much that I believe that people have a right to privacy as that I believe that people cannot possibly have a right to violate someone else's privacy.