If you dig into the Settings you can disable sharing chats with OpenAI, but you lose access to just about every feature including saving chats. You can only have one chat at a time, and if the window is closed or refreshed your chat is wiped. it's kind of like if opening a "Private Browsing" window prevented you from having a regular browsing window open and also had no tabs.
For some reason, they still retain your chat for "up to 30 days" despite not letting you save or access it after the page is refreshed.