Make sure the PC Subtitle Panel is assigned to the dialogue UI's StandardDialogueUI component > Subtitle Panels list and Default PC Subtitle Panel field.
As a test, can you temporarily assign the original Basic Standard Dialogue UI prefab to the Dialogue Manager's Display Settings > Dialogue UI field and check if the player's lines appear correctly? This will tell us whether the issue is with the dialogue UI or something else such as the dialogue database.
Would it be possible to send a reproduction project to tony (at) pixelcrushers.com?
Thank you for sending the reproduction project. I replied to your email. Please switch the Conversation Actor and Conversation Conversant in your Dialogue System Trigger.