Hi,
Network Monitor is in Applications/Utilities
However Activity Monitor in the same Folder is a better app for seeing if the process fires up when trying to Send.
IMTransferAgent should be the item you are looking for.

This shows it allowed in Little Snitch.
Little Snitch is a Firewall app that tell you when an app is making incoming or outgoing connections (if you have not set them up to be allowed)
It has to be allowed separately from Messages because the app does not see it as part of Messages. (the Mac Firewall does know and it is Allowed if Messages is allowed).
There are similar apps to Little Snitch that are Firewall app with reporting help.
In Sierra it is in:-
Where: /System/Library/PrivateFrameworks/IMTransferServices.framework/IMTransferAgent. app/Contents/MacOS/IMTransferAgent
In High Sierra the it is the same.
As they are in the same place it is strange why some people say it does not work.
However I don't have an issue so a poor Update could be missing parts of the IMTransferServices.framwork folder and/or contents.
If the Activity Monitor does not shoe the process starting then the Process in the above location may not exist.

9:01 pm Saturday; May 5, 2018
iMac 2.5Ghz i5 2011 (Sierra)
G4/1GhzDual MDD (Leopard 10.5.8)
MacBookPro 2Gb (Snow Leopard 10.6.8)
Mac OS X (10.6.8),
iPhone 6 iOS 11.x and an iPad (2)