You can see a history of your engagements in Dynamics 365. Each one includes a transcript of the conversation. This article explains how to find these in the Customer Service Hub. Similar features may be available in other apps, such as the Sales Hub.
To find these in the Customer Service Hub, go to the My Work heading on the left navigation panel and click on Activities.
This opens the Activities page. It is normally filtered to "My Activities," or the engagements you have been involved in.
To view all engagements, click the down arrow next to My Activities, then All Chat Activities.
This shows all engagement activities.
Clicking on a record shows details about the engagement.
Included on this page is a Chat Transcript URL. If you click on it, you can open the full transcript of the conversation between the agent and customer in a new window.
Important: Transcripts do not appear in this display during the conversation; they are added after a conversation has completed. If you have recently finished a conversation, you may need to click on the Refresh button to get the transcript to appear.
For details on how long they are stored, see Will all chat transcripts be stored within Dynamics 365?