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Of course they do, you can file a request so they would send you what info they hold. I did so. They store everything - your every message, all login dates, device metadata and anything else you can think of.
The best part? They say that the messages you send are like email, so they are not obliged to delete them.
Element as in the Matrix client? I'd be interested in this as well. A while back I reas that Matrix does like to phone home unnecessarily a lot even when you're self hosting it, it would be cool if that had changed (or was untrue).