Not sure about all inverters but when I fire up the generator, or am hooked up to shore power, the inverter still receives 120 volt power to power the charger portion and it is the inverter/charger that is charging the batteries. Depending on how the charger portion is programmed, it can draw quite a bit of current when charging the batteries. I also have pass-through current with my inverter when hooked to shore power or generator which can trip the inverter breakers stopping the 120 volt output to my outlets.
It may not be inverting but the inverter/charger still can draw a sizeable amount of power.