You should assume that chat logs can become public at anytime, if you host it yourself or not. Don't put sensitive/embarrassing unencrypted information in chat or email. People forget this is still data at rest.
A corollary of this is a saying a friend's dad used to have, "If it isn't in writing, it doesn't exist". Of course, with VOIP, video, etc, it is more, "if it isn't done in person via voice, it does exist". If it would be damaging to leak, seriously consider whether it should be written down at all, especially in casual conversations.