We present the results of a ~230 ks long X-ray observation of the relativistic double-pulsar system PSR J0737-3039 obtained with the XMM-Newton satellite in 2006 October. We confirm the detection in X-rays of pulsed emission from PSR J0737-3039A (PSR A), mostly ascribed to a soft non-thermal power-law component (photon index ~ 3.3) with a 0.2-3 keV luminosity of ~1.9E+30 erg/s (assuming a distance of 500 pc). For the first time, pulsed X-ray emission from PSR J0737-3039B (PSR B) is also detected in part of the orbit. This emission, consistent with thermal radiation with temperature kT=30 eV and a bolometric luminosity of ~1E+32 erg/s, is likely powered by heating of PSR Bs surface caused by PSR As wind. A hotter (~130 eV) and fainter (~5E+29 erg/s) thermal component, probably originating from back-falling particles heating polar caps of either PSR A or PSR B is also required by the data. No signs of X-ray emission from a bow-shock between PSR As wind and the interstellar medium or PSR Bs magnetosphere are present. The upper limit on the luminosity of such a shock component (~1E+29 erg/s) constrains the wind magnetization parameter sigma of PSR A to values greater than 1.
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