We investigate the interplay between prestress and mechanical properties in random elastic networks. To do this in a controlled fashion, we introduce an algorithm for creating random free-standing frames that support exactly one state of self-stress. By multiplying all the bond tensions in this state of self-stress by the same number-which with the appropriate normalization corresponds to the physical prestress inside the frame-we systematically evaluate the linear mechanical response of the frame as a function of prestress. After proving that the mechanical moduli of affinely deforming frames are rigorously independent of prestress, we turn to nonaffinely deforming frames. In such frames, prestress has a profound effect on linear response: not only can it change the values of the linear modulus-an effect we demonstrate to be related to a suppressive effect of prestress on nonaffinity-but prestresses also generically trigger a bistable mechanical response. Thus, prestress can be leveraged to both augment the mechanical response of network architectures on the fly, and to actuate finite deformations. These control modalities may be of use in the design of both novel responsive materials and soft actuators.