I am using the basic-auth twitter API (no longer available) to integrate twitter with my blog\'s commenting system. The problem with this and many other web APIs out there
You don't have to have a certificate on your server; it's up to the client whether they are willing to talk to an unauthenticated server. Key agreement can still be performed to establish a private channel. It wouldn't be safe to send private credentials to an unauthenticated server though, which is why you don't see SSL used this way in practice.
To answer your general question: you just send it. I think your real general question is: “How do I send sensitive data over an insecure channel—and keep it secure?” You can't.
It sounds like you've decided that security isn't worth the $10–20 per month a certificate would cost, and to protect Twitter passwords, that's probably true. So, why spend time to provide the illusion of security? Just make it clear to your users that their password will be sent in the clear and let them make their own choice.