The error is, when I click “Sync Now” on the MBP, the Activity panel starts, and appears to make some progress, but stops halfway (for ages), and then the log panel comes up with “Could not open the connection (The operation couldn’t be completed. Operation timed out)”.
Yes, both machines have file sharing on. Apple file sharing works fine both ways between them.
I think I found out what was wrong. One machine, the iMac, was connected to the network by way of a physical ethernet connection to the router, whilst the other was connected via Wi-Fi.
Despite the fact that the iMac was connected to the network via ethernet, I had to turn its Wi-Fi on and make sure it was connected to the network via Wi-Fi before it would sync.