Gyroscopes do not lose energy by the fact of their rotation. Consider spacecraft that are given a spin for gyroscopic stability. One spin and they keep going.
Gyroscopes lose energy from:
* friction in their bearings
* air friction from their rotating surfaces
* in the case of an electrically conductive rotor in a magnetic field, the subsequent eddy currents will result in heating of the rotor and corresponding loss of angular momentum
In all cases, heat is the final disposition of the energy lost.
When you spin a gyroscope up by hand, you are applying one force tangentially to the rim of the flywheel, but as you are holding the frame of the gyroscope, there is an opposing force applied at the bearings. This opposing force stops the gyroscope from being launched into the air, but because the two forces are acting at two different points of the flywheel, there is a turning moment on it. With the flywheel on bearings, this turning moment causes the it to rotate. These exact same forces are in play when you use a string to accelerate the flywheel - just that the radius of the applied force is smaller (decreasing the mechanical advantage, but increasing the maximum speed achievable) and the length of time it is applied is longer.