In more detail, the generator and shore power supply 120v to the transfer switch. The transfer switch transfers 110v to the inverter/charger. The charger unit internal to the inverter converts the 110v to 12+v and sends 12+v power to the house batteries, and that same 12+v power is sent on to the chassis (starter) batteries through the Big Boy/BIRD system after the house batteries have been sufficiently charged.
So... yes, both the generator and shore power will charge both the house and chassis batteries IF the transfer switch, inverter/charger, AND Big Boy/BIRD system are ALL working properly.
From the engine, the engine's alternator sends 12+v to the chassis batteries, and that same 12+v power is sent through the Big Boy/BIRD system after the chassis batteries have been sufficiently charged.
The Big Boy/BIRD system is an important system used to keep both the house and chassis batteries charged. Note: Some coaches have an ECHO charger or some similar bi-directional device instead of a Big Boy/BIRD system.
BIRD means Bi-directional Isolator Relay Delay.