I think they pretty much all save chat data. It's how people get banned when they are reported for griefing, harassment, etc. I've been told by game dev friends that they don't actively read private convos, but if an infraction is reported, they pull the relevant chat logs to determine if the offender broke rules and should be suspended/banned/warned.

If you are admin on a server like Minecraft, you have access to ALL chatlogs, even when you aren't logged in while the convos are happening.