I had the identical problem.
This finally worked: Even though I had connected BlueTooth between my iPhone and laptop, and connected LAN via my wifi system to the same router and LAN network, an ethernet connection between my laptop and the Wifi of the iPhone did not work together even though both were on the same TCP/IP LAN. I had to turn my laptop wifi on, which I connected to the same wifi and LAN as my iPhone, and then for the first time the "Continuity Camera" worked: I did Pages->Insert->Import from iPhone->Take Photo, then I waited for some moments (maybe 4 seconds (everything with Apple is slow)), then the Camera app on my iPhone started, and I took a picture. Then it had a "Use Picture" button, so I pressed that, and then the picture showed up in Pages.
I also had this same problem with Handoff before: I had to turn on the Wifi of my laptop in order to use Handoff with my iPhone. I don't like this since I consider wired more secure than wifi, but at least now I know the operational parameters of these features. It makes sense that if Contuinity Camera is a part of Handoff or uses similar principles that both don't work when using hardwire Ethernet.
I also confirmed the same behavior with my iMac: Continuity Camera works with iMac wifi on, does not work with iMac wifi off, even though hardwired to identical LAN. This time, I had my iPhone in dark (sleep mode, screen not lit), and it woke up and went to the photo app, so that part worked great.
iPhone 6+ with iOS 12.0.1
MacBook Pro (Retina, 15-inch, Mid 2015) with macOS Mojave Version 10.14
iMac (Retina 5K, 27-inch, Late 2014) with macOS Mojave Version 10.14
Wifi: UniFi AP-Pro Version 3.7.5.4969; Router: UniFi Security Gateway 3P 4.3.33.4936086
FEATURE REQUEST TO APPLE: MAKE HANDOFF (and Continuity Camera) WORK WITH WIFI OFF ON THE ETHERNET CONNECTED DEVICES (FOR INSTANCE, I USE ALL MY DEVICES, INCLUDING ETHERNET DEVICES, ON THE SAME LAN).
EVEN BETTER FEATURE REQUEST TO APPLE: MAKE HANDOFF (and Continuity Camera) WORK IF ANY CONNECTION TYPE CAN BE USED AT ALL (SUCH AS BLUETOOTH, LAN, INTERNET, ETC).