We present the results from a sensitive X-ray survey of 26 nearby hydrogen-poor superluminous supernovae (SLSNe-I) with Swift, Chandra, and XMM. This data set constrains the SLSN evolution from a few days until ~2000d after explosion, reaching a luminosity limit Lx~10^40^erg/s and revealing the presence of significant X-ray emission possibly associated with PTF 12dam. No SLSN-I is detected above Lx~10^41^erg/s, suggesting that the luminous X-ray emission Lx~10^45^erg/s associated with SCP 60F6 is not common among SLSNe-I. We constrain the presence of off-axis gamma-ray burst (GRB) jets, ionization breakouts from magnetar engines and the density in the sub-parsec environments of SLSNe-I through inverse Compton emission. The deepest limits rule out the weakest uncollimated GRB outflows, suggesting that if the similarity of SLSNe-I with GRB/SNe extends to their fastest ejecta, then SLSNe-I are either powered by energetic jets pointed far away from our line of sight ({theta}>30{deg}), or harbor failed jets that do not successfully break through the stellar envelope. Furthermore, if a magnetar central engine is responsible for the exceptional luminosity of SLSNe-I, our X-ray analysis favors large magnetic fields B>2x10^14^G and ejecta masses M_ej_>3M_{sun}_, in agreement with optical/UV studies. Finally, we constrain the pre-explosion mass-loss rate of stellar progenitors of SLSNe-I. For PTF 12dam we infer dM/dt<2x10^-5^M_{sun}_/yr, suggesting that the SN shock interaction with an extended circumstellar medium is unlikely to supply the main source of energy powering the optical transient and that some SLSN-I progenitors end their lives as compact stars surrounded by a low-density medium similar to long GRBs and type Ib/c SNe.