The Dialogue System Extras page now has an example scene that demonstrates how to set up simultaneous conversations. You can also download it here: SimultaneousConversations_2016-07-27.unitypackage
(If the link above no longer works, the package has been updated. Try the Extras page for the latest version.)
Both conversations play when the scene starts. An interactive conversation plays between Player and NPC1. Simultaneously, a bark conversation plays between NPC2 and NPC3.
This is how I set it up:
1. On the Dialogue Manager, I ticked Allow Simultaneous Conversations and set Subtitle Settings > Continue Button to Always.
2. On the main dialogue UI (under the Dialogue Manager), I ticked the Find Actor Overrides checkbox. It's at the bottom of the inspector for the Unity UI Dialogue UI component.
3. On NPC2 & NPC3, I added Generic Unity UI Bark UI prefabs as children.
4. On NPC2, I added Bark Dialogue UI and Override Display Settings components. On Override Display Settings, I set the Dialogue UI to NPC2 and Continue Button to Never.