under My Profile > Settings & Privacy > View the security center and adjust Two-Step Verification settings
It will take you here:
https://chaturbate.com/security/
Click "Privacy Center" which of course leads to here:
https://chaturbate.com/security/privacy/
Then "Export Data" which leads to:
https://chaturbate.com/security/privacy/data_export/
You can export all kinds of data by checking the boxes and choosing to Export.
One of the boxes you can check/tick is (third option down):
"Chat Messages
Chat messages sent by you".
They are, as it says, messages BY me sent to the model (and public chat I think).. I am wanting for messages (IN PM only, not bothered by public chat ones) sent by Models and received by me.