We tested whether joint spectrotemporal sensitivity follows from spectrotemporal separability for normal-hearing conditions and for impaired-hearing simulations. In a manual reaction-time task, normal-hearing listeners had to detect the onset of a ripple (with density between 0-8 cycles/octave and a fixed modulation depth of 50%), that moved up or down the log-frequency axis at constant velocity (between 0-64 Hz), in an otherwise-unmodulated broadband white-noise. Spectral and temporal modulations elicited band-pass filtered sensitivity characteristics, with fastest detection rates around 1 cycle/oct and 32 Hz for normal-hearing conditions. These results closely resemble data from other studies that typically used the modulation-depth threshold as a sensitivity measure for spectral-temporal modulations. To simulate hearing-impairment, stimuli were processed with a 6-channel cochlear-implant vocoder, and a hearing-aid simulation that introduced spectral smearing and low-pass filtering. Reaction times were always much slower compared to normal hearing, especially for the highest spectral densities. Binaural performance was predicted well by the benchmark race model of statistical facilitation of independent monaural channels. For the impaired-hearing simulations this implied a "best-of-both-worlds" principle in which the listeners relied on the hearing-aid ear to detect spectral modulations, and on the cochlear-implant ear for temporal-modulation detection. Although singular-value decomposition indicated that the joint spectrotemporal sensitivity matrix could be largely reconstructed from independent temporal and spectral sensitivity functions, in line with time-spectrum separability, a significant inseparable spectral-temporal interaction was present in all hearing conditions. These results imply that the reaction-time task yields a solid and effective objective measure of acoustic spectrotemporal modulation sensitivity, which may also be applicable to hearing-impaired individuals.