To Access ConversationEntry entity, “Access Conversation Entries” permission is mandatory.

To use the ConversationEntry object for Chat, enable the new Access Conversation Entries user permission.

Reference Link – https://help.salesforce.com/articleView?id=release-notes.rn_api_objects.htm&type=5&release=232

This is applicable to Chat, Messaging and Service Cloud Voice entities. 

If the user’s profile doesn’t have this permission, they will hit “You don’t have View access to Conversation Entries. Ask Salesforce Admin for access, then try again” exception.