Ok, more fun!
It seems that the topic does turn up via the API in /topics/private-messages-sent/username.json, but not in /topics/private-messages/username.json.
Also, if I go to the topic id URL directly at http://localhost:3000/t/neil-private-message/8695, it appears fine with the message icon, and the reply.
Am I perhaps missing something about where private messages should appear in the web UI or something?
Edit: Yes, yes I am stupid. It would help to click on ‘sent’ in the messages UI to view sent messages. 
I’ll go crawl into a hole now.