We log IP addresses and names in the chat. We also log any communication on the chat in order to allow super-moderators to review bans. These logs are kept for no longer than three months. We do not use these logs to actively search for rule violations, unless we have specific reasons (complaints, ban appeals, etc) to do so. Access to logs is restricted to a small number of moderators. You should not write or whisper sensitive information on the chat. Use secure communication channels (encrypted email or encrypted IM) for that. The chat is for casual conversation only.
As you can see, different parts of the site have different levels of logging. Part of this is due to technical circumstances, or administrative requirements. The main site uses as little logging as possible. The chat is very log-heavy, mainly due to the fact that moderators have to investigate complaints or ban appeals on a regular basis.
Always remember that we can not influence what data is stored by your ISP. Be aware of this fact if you're accessing SoFurry from within countries where such content is deemed immoral or illegal.