Can you typically see your services outside you network, maybe thru a web adapter? Also, do you use https, and/or are your services secure? Chances are you might need a proxy if they are secure.
In our situation, I have two web adapters set up, one for our unsecure services (mapping), the other for secure services (ags). I have a proxy with a token for the secure services and basically just a pass thru if the are not secure. Also, in my case I refer to all as https.
I typically refer to my ArcGIS Servers similar to this
https://<web URL>/<web adapter>/rest/services/<folder in AGS>/<servicename>/MapServer
So a few things are in play.....but the main thing is 1) they have to have some way to be seen outside the network (eg ports 80 or 443) and a proxy if needed.
It may help to know a few more specifics about your setup.
