We present physical properties [redshifts (z), star-formation rates (SFRs) and stellar masses (M*)] of bright (S_850_>=4mJy) submm galaxies in the ~=2deg^2^ COSMOS and UDS fields selected with SCUBA-2/JCMT. We complete the galaxy identification process for all (~=2000) S/N>=3.5 850-{mu}m sources, but focus our scientific analysis on a high-quality subsample of 651 S/N>=4 sources with complete multiwavelength coverage including 1.1-mm imaging. We check the reliability of our identifications, and the robustness of the SCUBA-2 fluxes by revisiting the recent ALMA follow-up of 29 sources in our sample. Considering >4mJy ALMA sources, our identification method has a completeness of ~=86 per cent with a reliability of ~=92 per cent, and only ~=15-20 per cent of sources are significantly affected by multiplicity (when a secondary component contributes >1/3 of the primary source flux). The impact of source blending on the 850-{mu}m source counts as determined with SCUBA-2 is modest; scaling the single-dish fluxes by ~=0.9 reproduces the ALMA source counts. For our final SCUBA-2 sample, we find median z=2.40^+0.10^_-0.04_, SFR=287+/-6M_{sun}_/yr and log(M*/M_{sun)_=11.12+/-0.02 (the latter for 349/651 sources with optical identifications). These properties clearly locate bright submm galaxies on the high-mass end of the 'main sequence' of star-forming galaxies out to z~= 6, suggesting that major mergers are not a dominant driver of the high-redshift submm-selected population. Their number densities are also consistent with the evolving galaxy stellar mass function. Hence, the submm galaxy population is as expected, albeit reproducing the evolution of the main sequence of star-forming galaxies remains a challenge for theoretical models/simulations.