The metal-semiconductor-metal (MSM) structure is a popular architecture for developing Ga2O3 solar-blind photodetectors. The nature of metal-semiconductor contact is decisive for the operation mode, gain mechanism, and device performances. In this contribution, κ-Ga2O3 MSM solar-blind photodetectors with Ti/Ga2O3 Ohmic and Ni/Ga2O3 Schottky contacts were constructed on the high-quality Si-doped κ-Ga2O3 epilayer grown by hydride vapor phase epitaxy. The Ti/κ-Ga2O3/Ti Ohmic MSM device is operated in a photoconductive mode, exhibiting a maximum responsivity of 322.5 A/W and a high rejection ratio of over 105, but with an undesirable sub-gap response and high dark current. In comparison, the Ni/Ga2O3/Ni photodiode with a back-to-back Schottky configuration is operated in a mixed photovoltaic and photoconductive mode, demonstrating a decent photoresponsivity of 0.37 A/W, a maintained high rejection ratio of 1.16×105, a detectivity of 3.51×1013 Jones and the elimination of slow photoresponse from sub-gap states. The frequency-dependent photoresponse and transient photocurrent characteristics indicate that the persistent photoconductivity effect is responsible for the high gain achieved in the Ti/Ga2O3/Ti photoconductor, and the dominant slow transient decay component is a fingerprint of photoexcited carrier trapping and repopulation. The response speed is improved in the Ni/Ga2O3/Ni Schottky MSM device, whereas carrier transport across interdigitated fingers is affected by bulk traps, limiting the overall response-bandwidth merit.