Nuclide production cross sections for proton-induced reactions on Mn and Co at incident energies of 1.3, 2.2, and 3.0 GeV were measured by the activation method at the Japan Proton Accelerator Research Complex (J-PARC). In total, 143 production cross sections of reaction products with half-lives ranging from 1.8 h to 2.6 yr were obtained. Among them, the cross sections of Mn-55(p,x) S-38, Mn-55(p,x)Ar-41, and Co-59(p,x)S-38 reactions were measured for the first time. The stable proton beam and well-established beam profile-motoring system contributed to the reduction of the systematic uncertainties to typically less than 5%, which was better than those of the previous data (typically similar to 10%). To examine the prediction capabilities of spallation reaction models and evaluated nuclear data library, the measured data were compared with the spallation reaction models in PHITS (INCL4.6, Bertini, and JAM as intranuclear cascade models, and GEM as an evaporation model), INCL++/ABLA07, and the JENDL/HE-2007 library. The comparison of the mean square deviation factors indicated that both INCL4.6/GEM and JENDL/HE-2007 showed better agreement with the measured data than the others.