Currently, there would no ways that you can chat using your twitter account via twitter official web page (twitter.com) between both of your twitter friends (follower / following). But don’t worry, there is another way to make that chat features for your twitter account enable.
Treegger – XMPP Server for your Twitter Account and Foursquare Account
By using Treegger in your XMPP / Jabber Client such as Adium, Psi, Imonair for Android / iPhone, iChat, Pidgin, Miranda, or others Jabber / XMPP Client you’ll be able to use your Twitter and Foursquare account.
The Default configuration that you should take for XMPP Client is like this one:
Jabber ID: [your twitter account name]@twitter (for example ‘jchasetg@twitter’) or your [foursquare email with '#' instead of '@']@foursquare (for example ‘jchase#treegger.com@foursquare’)
Connect Server: xmpp.treegger.com with port 5223
Allow plaintext authentication (twitter doesn’t provide any other authentication method yet)
SSL/TLS (old style) required and selected
Remember you should make sure to check the SSL / TLS option to make sure this chat via treegger.com server works properly. It because the treegger would guarantee that your data messages (chat text) runs through the server securely (encrypted).