Does a Target Service need to do anything for a conversation that is ended? We have two queues, SSBInitiator and SSBTarget.  In my example situation, I have ran the following code to create a conversation:

BEGIN DIALOG CONVERSATION @ConversationHandle
                FROM SERVICE [//BLAH/Events/SSBInitiator]
                TO SERVICE '//BLAH/Events/SSBTarget'
                ON CONTRACT [//BLAH/Events/SSBContract]
                WITH ENCRYPTION = OFF;


Now lets say that, for some reason I've decided this conversation is unnecessary: either the user cancelled the request, or something else happened.  I know I can end the conversation using either:

end conversation @ConversationHandle


or

END CONVERSATION @ConversationHandle   
      WITH ERROR = 1 DESCRIPTION = 'Ending Convo cause reasons';


Now, with the conversation ended and no messages have been sent on it, does the Target service need to do anything with the conversation?  IE, is there additional processing that needs to happen to make sure the conversation is completely closed?
From Microsoft Docs:

Both participants in a conversation must call END CONVERSATION for the conversation to complete.

If only one side ends the conversation, then the conversation metadata stays valid.
The END CONVERSATION ... WITH CLEANUP can be used to clean up any un-ended conversations.
Remus Rusanu has an excellent article showing how to do that here.