Protostars interact with their surroundings through jets and winds impacting on the envelope and creating shocks, but the nature of these shocks is still poorly understood. Our aim is to survey far-infrared molecular line emission from a uniform and significant sample of deeply-embedded low-mass young stellar objects in order to characterize shocks and the possible role of ultraviolet radiation in the immediate protostellar environment. Herschel/PACS spectral maps of 22 objects in the Perseus molecular cloud were obtained as part of the `William Herschel Line Legacy' survey. Line emission from H$_\mathrm{2}$O, CO, and OH is tested against shock models from the literature. Observed line ratios are remarkably similar and do not show variations with source physical parameters. Observations show good agreement with the shock models when line ratios of the same species are compared. Ratios of various H$_\mathrm{2}$O lines provide a particularly good diagnostic of pre-shock gas densities, $n_\mathrm{H}\sim10^{5}$ cm$^{-3}$, in agreement with typical densities obtained from observations of the post-shock gas. The corresponding shock velocities, obtained from comparison with CO line ratios, are above 20 km\,s$^{-1}$. However, the observations consistently show one-to-two orders of magnitude lower H$_\mathrm{2}$O-to-CO and H$_\mathrm{2}$O-to-OH line ratios than predicted by the existing shock models. The overestimated model H$_\mathrm{2}$O fluxes are most likely caused by an overabundance of H$_\mathrm{2}$O in the models since the excitation is well-reproduced. Illumination of the shocked material by ultraviolet photons produced either in the star-disk system or, more locally, in the shock, would decrease the H$_\mathrm{2}$O abundances and reconcile the models with observations. Detections of hot H$_\mathrm{2}$O and strong OH lines support this scenario.