I think you're not looking at the complete picture here. Let me explain:

Lots of people are giving out their personal information right now in exchange for basic data sharing with others. In most cases (and this is specially true for people who aren't computer experts--i.e., most users) users are a) not even aware that they are doing it, and b) without any reasonable choices other than not sharing their data at all.

It's hard to find good easy-to-understand examples, but one analogy would be if all of the shipping companies kept a file of you and indexed every letter you'd ever sent. You get to choose whether you send mail to a friend or whether your data stays private.

Now, what is different with Weave? Weave encrypts all data placed on the server before putting it there. Only you have the passphrase that unlocks your data. Sharing can be implemented by means of public-key encryption, where you can securely give access to a 3rd party while keeping the data encrypted from everyone else. And without sharing your username, password, or encryption passphrase with anyone (including the 3rd party).

Of course, some people don't care about privacy, and will allow their data to be public. And that's fine too--as long as it's a user decision and not just the lay of the land.