Posting messages to your public Twitter feed means two things for your privacy. First, your followers can see the message immediately (and sometimes even if you later delete it) and second, third party applications may archive and display your feed for years to come.
You can protect your privacy by logging into Twitter and changing your privacy settings with the Settings button found in the upper right corner of the screen. Under the Account tab, select Protect My Updates. Protecting your updates removes them from the public timeline and keeps search engines from indexing them.
This is, however, not a complete solution. Anything you posted before making your feed private will likely remain public. Additionally, any follower who you allow to see your updates might “retweet” your tweets, which could make them public. In a more extreme scenario, if Twitter was hacked, your messages could be compromised.
In short, there is never a guarantee that your Twitter messages will stay private. You should never rely on the Twitter privacy settings to protect your privacy – the most effective thing you can do is to take care to not post private or sensitive details on Twitter.