(abridged:) We study X-rays from jet-driving protostars and T Tau stars. We seek soft spectral components that may be due to shock emission, and shock-induced emission displaced from the stellar position. Two stellar samples are used, the first consisting of lightly absorbed T Tau stars with strong jets, the other containing protostars with disks seen nearly edge-on. The former sample was observed in the XMM-Newton Extended Survey of the Taurus Molecular Cloud (XEST), while Chandra archival data provided observations of the latter. We confirm the previously identified peculiar spectrum of DG Tau A and find similar phenomenology in GV Tau and DP Tau, suggesting a new class of X-ray spectra. These consist of a lightly absorbed, very soft component and a strongly absorbed, very hard component. The latter is flaring while little variability is detected in the former. The absorption of the harder component is about an order of magnitude higher than expected from the optical extinction assuming a standard gas-to-dust mass ratio. The flaring hard component represents active coronal emission. Its strong absorption is attributed to mass inflow from the accretion disk. The optical extinction is small because the dust has sublimated at larger distances. The weakly absorbed soft component cannot originate from the same location. Because the stars drive strong jets, we propose that the X-rays are generated in shocks in the jets. We find that for the three peculiar X-ray sources, the luminosity of the soft component roughly scales with the equivalent width of the [OI] 6300A line formed in the jets, and with the mass outflow rate. In the more strongly obscured protostars, the soft component is entirely absorbed, and only the hard, coronal component penetrates the envelope or the near-edge-on disk.