I would strongly recommend that you store your chat history since a simple search for " Chat History in Google" you see that its a feature a lot of people request for in chat clients.
It would be amazing if you could have all of your chat history(and
other history such as emails, file transfers, tweets, etc) all saved
online so that when you go and use another computer, or you reformat
your computer, or in some other way lose access to your locally stored
logs, you can still access them. Similar to how the Meebo
client(www.meebo.com) does it.
Also it seemed to be a very highly requested feature in google chat since if you look at the chat history FAQ for google, you see this text
One of our users' most commonly-requested features is finally here!
Now you can save your chat histories to your Gmail account. There,
your chat histories are just like Gmail messages--you can view them,
forward them, print them or search for them. Never worry about losing
something important from your chats again.
Lastly from a personal front, I would prefer that chat history be saved since lot of useful information can be shared during chat communications (email ids, phone numbers, web urls,addresses etc). I know you do have the feature to send messages but it would be really bad UX if your user has to specifically type out a message to send useful information when it could have been sent as part of a chat conversation.
Another reason you should save chat histories is that it serves as a reference to what the previous conversation was about and can be a starting point about previous topics. eg:
Person 1 : Hey,how are you feeling now ?
Person 2: I am fine, why what happened ?
Person 1 : Oh,I was just looking at our chat logs and the last time we chatted, you had a pretty nasty cold
Note : Though I strongly recommend the ability to save chat history, always ensure the user has a way out of it,i.e. he can disable the chat history from being saved. Also ensure the user can delete specific chat histories or even all the history when ever he wants to.
However I would also recommend running some usability tests to see what users have to say about the feature and they find it useful or intrusive with regards to privacy