This question comes up a lot regarding different heating appliances. There are lots of advertisements for small heating devices that have extravagant claims of efficiency, effectiveness and power! As far as I know, 1,000 watts is 1,000 watts. That means any resistive heating device no matter how large, long or small that is rated 1,000 watts should produce the same amount of BTUs. This means the heater the Amish make and sell in the Parade Magazine produces the same heat as any other similar device.
If this is not the case I must be missing something?
You are correct, watts is watts.
Those heaters you mention, as well as many others sell very well, because people do not understand the physics. I the infomercials you see they always have people mentioning "no heating bill" or "reduced heating bill". If your homes heating system is gas, of course, the gas bill will go down, you used less gas, your electric bill should have gone up though. Now if you are using the heater in only one room, you very well do still save some money, as you are only heating that room, or maybe you leave the rest of the home at a lower temp and only raise it in a single room where you occupy the most. By doing that you are "zone heating". Any electric heater will use the same in energy to heat that room to the same temperature, it does not have to be the one they are trying to sell you.
If you have a resistance electric heating system in your home, you may save some with "zoning" certain rooms and keeping others at lower temp, but if you have one of their units in every room and set each to same temp, you will save nothing, it should take same energy from your central system to do the same thing.
If you have gas central system and you put one of their heaters in every room, all set to same temperature, you will only save on energy cost if electricity cost is less than gas cost. This will not be true for most of the country. Couple years ago gas was high enough that many might have saved some money, today gas is lower and that may not be the case. I have not looked into specific rates recently, but have heard gas is much lower than it was a couple years ago in most places.