The link provided by the now retired LogMeIn contributer is no longer valid.
We discovered that our chats were being saved to our Documents folder (this is a PC).
We are not sure if this is automatic, or if this was something we set up a long time ago.
I am not finding this preference setting anywhere: a preference that can be set to automatically save the chat log to your computer.
Alright, I now see the Preferences area when I right-click on the GoToTraining icon in the system tray.
I had been launching the My Trainings interface in GoToTraining to get to the admin view, then clicking on the Settings tab, which had very limited options.