Most of the generators I come across have the coolant circulation style heater. There are still some that use a plug style but those are likely to have something like this occur:
The coolant circulating style heat the coolant so it starts boiling and rising up causing the coolant to circulate through the engine. For most units it only circulates through the engine and not the rad because if your radiator is working properly the coolant will be cold again by the time it gets to the bottom. This constant circulation allows the entire engine block to warm up somewhat evenly. If it fails to circulate the block heater will burn out. This is actually quite common and block heaters do fail somewhat regularly. Many people think that an indoor generator doesn't require a block heater because it's never cold. The block heater doesn't just warm up the oil it also makes it easier for the engine to start. The aluminum pistons warm up much quicker than the engine block. A block heater eliminates any period of time that the pistons have expanded before the block causing them to scuff against the cylinder liners.
Here is a picture of a typical coolant heater. You can see that the element is inside of the plastic piece with cold coolant entering from the bottom and hot coolant boiling up the top.
Many people seem to believe that only diesel engines use this style of heater but they are even common on small gas engines like this little four cylinder Datsun:
The block heater is the most common area for coolant leaks. You will see the residue left on the hoses or the heater itself like this:
These leaks are easy to spot and usually very easy to fix.
The most common problem I have with block heaters is that they attract rodents! A toasty warm engine is like a luxury hotel to mice so make sure you animal proof the enclosure!