Welcome! It’s a good question. It depends on the specific brand and model of the switch, as well as how you have installed it.
The Master Switch
A “master” switch is generally installed as a load controlling switch, which means if the Internet is off or if the SmartThings cloud is unavailable or if your smartthings hub is removed altogether, The switch still works just like a regular dumb switch would work. Turn it on and the light fixture comes on. Turn it off and the light fixture goes off. However, that only applies to using it just like a dumb switch and physically using the switch on the wall. You won’t be able to use the phone app to control the switch.
Any special features, like a double tap to turn on a light in another part of the room, or the single tap normally turning on several lights on several circuits might or might not work, but probably won’t work.
An auxiliary switch in a three-way lighting set up, like one Switch at each end of the hallway that both control the same light
In the set up, the exact model of the devices matters a lot. There are some devices, specifically those that use physical traveler wires to communicate to the master switch, which will still do that basic on/off even if none of the home automation system is working.
But there are other models which use a “virtual 3 way,” where the auxiliary communicates to either the master or the smartthings hub wirelessly rather than directly through wires. Again, depending on the exact set up, those might or might not work if the home automation system wasn’t working. But probably they won’t work. The master will still work. So in that hallway example, it might mean that only one of the two switches worked if the home automation system was down.
If making sure that every auxiliary switch Will still have on/off capability at the Switch if the home automation system is down is important to you, you need to select models that have that capability, as not all do.
Wallmount remotes, including any battery operated devices and switches that were wired to bypass the load
Many people use various kinds of wall mount remotes, including multibutton panels, tablets with fancy dashboard apps, individual buttons, etc. Those can be a real convenience factor.
In this case, it’s generally the same as the wired auxiliary switches. There are a few models which might still be able to work even if the Internet was down or the hub was missing, but not many, and you need to look specifically for that feature if you want it. In general, the wallmount remotes won’t work unless the full home automation system is working.
Master switches are typically wired so that they will function just like a regular dumb switch if the hub or the cloud or not available. But only the basic functionality for that single circuit will operate in these conditions. Fancy features like double tap and multi circuit control usually won’t work.
Other devices, Including mains powered auxiliaries and battery powered devices typically won’t work. There are a few exceptions, but if you want that feature, you need to look for it specifically when you select devices.
Hope that helps.