I was trying for a few days to get ARD to work on a remote computer running macOS Mojave, any other OS was fine just not Mojave. I found a link on the web suggesting it doesn't work as it used to and might require user(s) to open up screen sharing for you. That of course doesn't work when you are supporting many computers remotely.
What I was able to do to fix my problem was to VPN into the remote network, SSH into the computer and modify these 2 plists:
using my computers plists as a reference. Once they were modified I moved them back to the remote computer and then ran a repair on disk permissions. Rebooted the computer remotely and it worked for me. I certainly hope this is a fix (a very unnecessary fix I must say). Give it a go.