We present a 2D study of star formation at kpc and sub-kpc scales of a sample of local (z<0.1) U/LIRGs, based on near-IR VLT-SINFONI observations. We obtained integrated measurements of the star formation rate (SFR) and star formation rate surface density, together with their 2D distributions, based on Br_gamma and Pa_alpha emission. We observe a tight linear correlation between the SFR derived from our extinction-corrected measurements and that derived from 24 micron data, and a reasonable agreement with SFR derived from total IR luminosity. Our near-IR measurements are on average a factor 3 larger than optical SFR, even when extinction corrections are applied. We found that LIRGs have a median-observed star formation rate surface density of 1.72M_{sun}_/yr/kpc^2^ for the extinction-corrected distribution, whilst ULIRGs have 0.23M_{sun}_/yr/kpc^2, respectively. These median values for ULIRGs increase up to 2.90M_{sun}_/yr/kpc^2^, when only their inner regions, covering the same size as the average FoV of LIRGs, are considered. We identified a total of 95 individual SF clumps in our sample, with sizes within 60-1500pc, and extinction-corrected Pa_alpha luminosities of 10^5^-10^8^L_{sun}_. Star-forming clumps in LIRGs are about ten times larger and thousands of times more luminous than typical clumps in spiral galaxies. Clumps in ULIRGs have sizes similar (x0.5-1) to those of high-z clumps, having Pa_alpha luminosities similar to some high-z clumps, and about 10 times less luminous than the most luminous high-z clumps identified so far. We also observed a change in the slope of the L-r relation. A likely explanation is that most luminous galaxies are interacting and merging, and therefore their size represents a combination of the distribution of the star-forming clumps within each galaxy in the system plus the effect of the projected distance.