For a simple heater with only one output power setting, it is actually just a switch. For 110V, two 110V heating elements in parallel will produce the required amount of power. For 220V, switching the same two 110V elements so they are in series will produce the same power from the same elements.Which is how I came to be thinking that if you turn the switch on the device to 110v, maybe that means it'll do something internally (like cut off half the heating element) so it only draws half the current/watts/amps/joules/kangaroos or whatever...
A real hair dryer will have several heat settings for each voltage. but the same principle applies.