We carry out a systematic study of the X-ray emission from the active nuclei of the 0.02<z<0.7 2Jy sample, using Chandra and XMM-Newton observations. We combine our results with those from mid-infrared, optical emission-line and radio observations, and add them to those of the 3CRR sources. We show that the low-excitation objects in our samples show signs of radiatively inefficient accretion. We study the effect of the jet-related emission on the various luminosities, confirming that it is the main source of soft X-ray emission for our sources. We also find strong correlations between the accretion-related luminosities, and identify several sources whose optical classification is incompatible with their accretion properties. We derive the bolometric and jet kinetic luminosities for the samples and find a difference in the total Eddington rate between the low- and high-excitation populations, with the former peaking at ~1% and the latter at ~20% Eddington. Our results are consistent with a simple Eddington switch when the effects of environment on radio luminosity and black hole mass calculations are considered. The apparent independence of jet kinetic power and radiative luminosity in the high-excitation population in our plots supports a model in which jet production and radiatively efficient accretion are not strongly correlated in high-excitation objects, though they have a common underlying mechanism.