I'm sorry, I really have no idea what you're asking.
The workflow of the plugin is:
1. You specify specify the admin email to receive responses
2. You enable "Debug Mode" -- these will be sent to #2
3. You create a service and attach it to specific forms
4. The service will submit to an indicated url; let's say you use the included 'service_test.php'
5. You set up service mapping:
6. Someone submits the attached form
7. The contact form plugin (CF7 or GF) behaves regularly, sending confirmation emails, printing a message to the screen, etc
8. The 3rdparty-Integration plugin will, at the same time, perform a form POST to the service url from #4, using the mappings from #5 to turn the contact form plugin submission into a service request
9. If the Integration plugin encounters an error with the service request (i.e. a complete failure of the response, or the lack of the specified "success" text) it will send a failure notice to the admin email, and it may update the contact form plugin notification message with a generic "could not send the reply" type message; in the case of CF7 it will print a developer-tools console warning, but for GF it should append to the msg shown on screen or include in the redirect url.
10. Regardless of success or failure, because you enabled Debug Mode in #2, you will get an email detailing the service, the submission, the mapped request, and the response.
After re-reading what I just wrote, maybe you mean in #9 it should have displayed a failure message but did not -- is that what you're talking about?