We present XMM-Newton X-ray observations of the interacting galaxy pairs NGC 7771 7770 and NGC 23422341. In NGC 7771, for the first time we are able to resolve the X-ray emission into a bright central source plus two bright (L-X > 10(40) erg s(-1)) ultraluminous X-ray sources(ULXs) located either end of the bar. In the bright central source (L-X similar to 10(41) erg s(-1)), the soft emission is well-modelled by a two-temperature thermal plasma with kT = 0.4/0.7 keV. The hard emission is modelled with a flat absorbed power-law (Gamma similar to 1.7, N-H similar to10(22) cm(-2)), and this together with a low-significance (1.7sigma) similar to300 eV equivalent width emission line at similar to6 keV are the first indications that NGC 7771 may host a low-luminosity AGN. For the bar ULXs, a power-law fit to X-1 is improved at the 2.5sigma level with the addition of a thermal plasma component (kT similar to 0.3 keV), while X-2 is improved only at the 1.3sigma level with the addition of a disc blackbody component with T-in similar to 0.2 keV. Both sources are variable on short time-scales implying that their emission is dominated by single accreting X-ray binaries (XRBs). The three remaining galaxies, NGC 7770, NGC 2342 and NGC 2341, have observed X-ray luminosities of 0.2, 1.8 and 0.9 x 10(41) erg s(-1), respectively (0.3- 10 keV). Their integrated spectra are also well-modelled by multi-temperature thermal plasma components with kT = 0.2- 0.7 keV, plus power-law continua with slopes of Gamma = 1.8-2.3 that are likely to represent the integrated emission of populations of XRBs as observed in other nearby merger systems. A comparison with other isolated, interacting and merging systems shows that all four galaxies follow the established correlations for starburst galaxies between X-ray, far-infrared and radio luminosities, demonstrating that their X-ray outputs are dominated by their starburst components.