David's answer is valid for iTalki classroom. And if the communication tool was Skype, you can also access access the chat there easily. But when you use Zoom, you have to save the chat before you leave the meeting or else you won't be able to retrieve it later.

Howdy Mary. The chat from lessons is not stored in a lesson-specific list, rather the messages exchanged during lessons are stored in the “messages tab”. So if you’ve had a lesson with Teacher X, say, in the messages tab you should see a selection marked “Teacher X”. Clicking on that will pull up all the messages you’ve exchanged with each other, both during and outside of lessons. The message list logs everybody you exchange messages with, starting with the most recent.