This Letter analyzes photoconductive (PC) terahertz (THz) emitters based on the semi-insulating (SI) forms of GaAs and InP. The dependencies of the emitters are studied under the extremes of the bias field and pump fluence to reveal the underlying physics of charge carrier photoexcitation, transport, and emission. The bias field dependence shows that SI-GaAs PC THz emitters are preferentially subject to space-charge-limited current, under the influence of trap states, while SI-InP PC THz emitters are preferentially subject to sustained current, due to a prolonged charge carrier lifetime and the ensuing joule heating. The pump fluence dependence shows space-charge and near-field screening for all emitters, with SI-GaAs predisposed to near-field screening (under the influence of transient mobility) and SI-InP predisposed to space-charge screening. Such findings can support a deeper understanding of the underlying physics and optimal performance of SI-GaAs and SI-InP PC THz emitters.