I take actual conversations from chat (private conversations are no exception!), remove the useless bullshit and distractions, and then format what remains.
I replace the user's names with emoticons (whatever I feel best matches the user during the conversation). I am always the ":3" face, obviously. I will also replace users names when used within the conversation, when I do this, I will use parentheses to signify when a user is referring to another user in the conversation.
I color code also. Using whatever color I feel matches the user at the moment. Except me, I am always green.