I get twitter.
And I think I get that you've added it.

But, it would seem that there is no way to configure your own twitter account in your profile, or have I missed it?

I see where to configure one for your forums, but that's only good for the owner.

Am I missing something?

Or does it rely on you being logged into Twitter in your local browser already?

The tweet buttons reply on the user to click it and posts it to their twitter linking to the post.

You can create a twitter page for your forum, people subscribe to that twitter page. You feed your twitter page stuff from your forum Same as Facebook