You could use two transmissions but they should be sync' together somehow so as to change gears at the same time. Running through one tranny was also done, generally with one engine in front connected to the front of the crank shaft of the rear engine with a flex joint. (Generally two sprockets side by side connected by a double row chain wrapped around both sprockets)
The one transmission thing is a more logical design because it simplifies things while the torque buildup on a SLR car is slow enough to allow a race prepared automatic to put up with the additional horsepower of two engines.
Many SLR cars only have two speeds anyway.
As for two differentials.... I don't think that was ever done! It couldn't work unless one gear set was removed. You only need one set anyway, so a show-car like the Invader probably had two rear axle tubes welded together with one differential gear set allowing for the slippage between left and right wheels while the other differential was solid.
Many early 60's Dragsters didn't even use a transmission. Most just had a clutch and an in-and-out box for idling at the line.
A couple of early dragsters used two axles to solve the problem of coupling by running one narrowed axle behind the wider wheel set. This allowed the axles to be mounted very close together. ("Jack the Bear" is what I think it was called) The builder later said it would have worked better if the two axles had been mounted on a common "Bogie" pivot. Unfortunately, the way it was built, the front end tended to lift when it accelerated which lifted the forward drive axle slightly unloading the wheels which would spin even faster creating a huge amount of rubber smoke....