As King Ghidorah rises out of the ocean after being revived for the second time, Godzilla also rises to the surface. We can hear Godzilla roaring, but you can see in the background here that Godzilla's mouth is not open to roar at all.
When Mothra is on Godzilla's head and wrestling with him, just before King Ghidorah first arrives, you can briefly see some sort of black rigging protruding from Mothra's back, most likely a device used to support the Mothra puppet.
The unnatural way King Ghidorah's two outer heads attach to his body, and the way the move, clearly reveal there's an actor inside his suit, and that these two heads are actually arm puppets. In other films, the heads were moved by strings, producing a much less fake-looking effect.