They wouldn't be using electricity to heat the water. Of course you would end up using less gas if you just heated the water directly like most of us already d

They wouldn't be using electricity to heat the water. Of course you would end up using less gas if you just heated the water directly like most of us already do for our morning shower. The efficiency of any heat engine depends on the gradient between the input and output temperature. So if you have a constant supply of cold water the Stirling engine can operate very efficiently, and the cold water becomes hot water which you can use for other things. Of course, like I said, this is true of any heat engine. You can do the same thing by heating a boiler and using the steam to drive a turbine, which also generates waste heat for other things.