Your name and a link to your Facebook profile is attached to every single Facebook chat you take place in; any user with access to your conversation log can trace it back to your profile page. Whenever you have a conversation on Facebook Chat, it gets logged with your messages on your Facebook account; this occurs for every other user in the chat. Your Facebook chat logs never display publicly on your profile, nor do they appear in searches.
Facebook chat conversations can only be initiated between confirmed friends. Facebook has no such thing as anonymous chatting. Your full name as entered in your Facebook profile will display, and can be clicked on by the person you're chatting with in order to go to your page. If the other member of a conversation chooses to forward your messages to another Facebook member, that conversation can be easily traced back to you as a participant -- however, the third party will only be able to view data on your profile according to your privacy settings. If you have your website, e-mail address or instant messaging username displayed on your profile, that information can be used in a Google search for users trying to get more information about you.
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During the course of a conversation on Facebook you can read your messages in the chat window; sometimes these messages will appear in the chat window next time you open the window. You can clear the window by clicking the gear icon on the window title bar and selecting "Clear Window." Clearing the window will not delete the chat from your messages log. You can only delete your copy of chat logs; they still remain logged for other users involved in the chat.
Deleting Chat History
To permanently delete a conversation in Facebook, you'll need to do so from the Messages interface. In the conversation window click the gear icon in the upper right corner and select "See Full Conversation." Click the "Actions" button and select "Delete Messages." You can choose select messages to remove from the chat log, or choose to delete the whole log. If selecting individual messages, click the "Delete Selected" button; you can delete everything by clicking "Delete All."
Users can access their Facebook chat via third-party clients like Pidgin, iChat, Trillian, AIM and any other chat client that allows users to add accounts using Jabber or XMPP. These instant messaging clients can be set to log every conversation; even if you delete the conversation on your account, these logs will still exist on the other user's computer. However, these conversations do not link to your Facebook profile in the same way as conversations that use the Facebook interface; these logs cannot be as easily traced back to you.