Consider:

[hendry@sg ~]$ hostname
sg.dabase.com
[hendry@sg ~]$ logout
Connection to 128.199.115.232 closed.
X1C3:~$ hostname
X1C3

sg is the name of my server. To address it from anywhere on the internet, it’s simply sg.dabase.com.

X1C3 is my laptop. It’s not online all the time. It’s usually on a 192.168.x.x address inside a NAT. To address it in a LAN it’s simply X1C3.local since all the devices on my LAN run Avahi/Zeroconf/Bonjour/mDNS hostname resolution. You cannot connect to it from anywhere on the Internet. However if you are in the same LAN/vicinity/network as I am, you should be able to explore my Web apps off http://X1C3.local/.

Issues

How to test hostname if it’s a remote or local machine?

In shell it could be:

if test $(hostname) == $(hostname | cut -d. -f1); then echo local; else echo remote; fi

How to test mDNS is running correctly in “private infrastructure environments”?

In shell it could be:

ping -c 1 $(hostname).local &>/dev/null && echo mDNS is working