I've found that you have to click the configure button (wrench) on the slide out bar, type in a password (it doesn't seem to matter what you type here) and then it will spawn a dialog actually asking for the password, after which point it will work.

Is it possible I have set this up wrong? This working for anybody else? Is there another plasmoid I ought to try for reading tweets? I don't care about posting them, I just want to see the latest ones on my desktop.

You may need to enable KWallet infrastructure in System Settings > Advanced > KWallet which will allow it to save your passwords. It can also be used by other KDE applications such as KMail and Konqueror in order to protect your passwords. You will have to provide your master password on login however.