I could be wrong, but I don't believe there are any other ways to read the transcripts outside of the reporting section.  At least, I've never seen any other way.

LayerZeroIssue is correct - outside of any integrations, the only place to check a chat transcript is through the reporting pages within /login. You can grant reps the ability to see only their own sessions/reporting data as well to allow this.

With some integrations, you may be able to view session data outside of this, like through SNOW, Outbound Emails, etc.

In theory you could probably write a utility with the API to do this. I might end up doing that if there is enough demand from the main technicians. 

Thanks!