Does the Personal Hotspot work when traveling?

Hotspot will work as long as you have a cellular network working in your device. It cannot work connected to wifi.

With regards to battery usage, it's yet another service enabled on your device, so it will use more battery than when it is disabled. I have not tested it, so I cannot say how fast is that depletion.

How to set up a Personal Hotspot on your iPhone or iPad - Apple Support

When you‘re using personal hotspot, your cellular radio is running at full power to maintain a constant cellular data connection. And your WiFi radio is at full power to maintain the WiFi connection to whatever you have using your hotspot. Ordinarily for internet access, your device by itself would only be connecting over cellular, or WiFi but would not be powering both simultaneously. So yes, using hotspot connections is a heavy power usage feature.

Note that when you’re traveling outside of your native carrier’s coverage area, hotspot will be using roaming cellular data connections if you have not purchased a travel plan from your carrier or are not using a less expensive local carrier.