Does spinning two generators at once, compared to one, generate more electricity? Let there be a turbine spinning a generator, as usual in our daily lives.
Assume that it is possible to connect another generator on the other side, as shown in the picture below.

Is it true that the amount of electricity generated is twice the original amount?
 A: The electrical power output of the generators cannot exceed the mechanical power input from the turbine (law of conservation of energy).  Thus if you add another  generator and want a greater electrical power output the mechanical energy input from the turbine must be increased.
A: 
Is it true that the amount of electricity generated is twice the original amount?

No. If you don't change anything on how the generator is driven mechanically, it will just spin slower. You don't get more energy out.
A: As a first-order approximation, the voltage at the terminals of an electrical generator will be proportional to the rotation rate of the shaft, and the current (Amperes) that it generates will be proportional to the torque applied to the shaft.
Most electrical and electronic devices expect a constant voltage power supply. If you want your two generators to generate the same voltage as in the single generator setup, then the turbine must spin at the same rate in either case.
You said, "amount of electricity." What matters in the economics of electric supply is the power (Watts). Power is the amount of energy delivered per unit time. The relationship between Volts, Watts, and Amperes is simple:
$P=VI$
Power output in Watts ($P$) equals the current in Amperes ($I$) times  the Voltage ($V$).
That looks a lot like the formula for the power in a shaft:
$P=\omega{}\tau{}$
Power in Watts ($P$) equals the rotation rate in radians per second ($\omega$) times the torque in Newton Meters ($\tau$).
You can't get power for free.
If you hope to generate twice as much power by using two generators, and assuming you want the same voltage, then the two generator setup will have to produce twice as much current. Twice as much current means, the turbine will have to produce twice as much torque as before. (I.E., it will have to work twice as hard as before to turn the shaft.)
If the turbine is unable to produce that much torque, then it will slow down, and the voltage will drop, and your electrically powered appliances will not work.
