We present XMM-Newton EPIC observations of the two nearby starburst merger galaxies NGC 3256 & NGC 3310. The broad-band (0.3-10 keV) integrated X-ray emission from both galaxies shows evidence of multi-phase thermal plasmas plus an underlying hard non-thermal power-law continuum. NGC 3256 is well-fit with a model comprising two MEKAL components (kT=0.6/0.9 keV) plus a hard power-law (Gamma=2), while NGC 3310 has cooler MEKAL components (kT=0.3/0.6 keV) and a harder power-law tail (Gamma=1.8). Chandra observations of these galaxies both reveal the presence of numerous discrete sources embedded in the diffuse emission, which dominate the emission above ~2 keV and are likely to be the source of the power-law emission. The thermal components show a trend of increasing absorption with higher temperature, suggesting that the hottest plasmas arise from supernova-heated gas within the disks of the galaxies, while the cooler components arise from outflowing galactic winds interacting with the ambient interstellar medium (ISM). We find no strong evidence for an active galactic nucleus (AGN) in either galaxy.