I don't get it. Twitter is a one to many broadcast channel. Its point is to be public. Privacy doesnt factor in. I think you could make a fake email, but twitter serves as a public channel so if you want privacy and twitter you are sort of asking for an sms message?

You give up privacy in many ways on Twitter, not just when you publish content.

Twitter can track you across third-party sites using its widgets, and it also knows your subscribing/reading behavior.

A decentralized system could be self-hosted and only pull in data through RSS, keeping you as anonymous as the IP address that you use. You'd only give up anonymity by replying to posts.