We have identified 7824 radio sources from the 1.4GHz NRAO VLA Sky Survey (NVSS, Cat. <VIII/65>) with galaxies brighter than K=12.75mag in the Second Incremental Data Release of the 6 degree Field Galaxy Survey (6dFGS DR2, Cat. <VII/249>). The resulting sample of redshifts and optical spectra for radio sources over an effective sky area of 7076deg^2^ (about 17 per cent of the celestial sphere) is the largest of its kind ever obtained. NVSS radio sources associated with galaxies in the 6dFGS span a redshift range 0.003<z<0.3 and have median z{bar}=0.043. Through visual examination of 6dF spectra we have identified the dominant mechanism for radio emission from each galaxy. 60 per cent are fuelled by star formation and 40 per cent are fuelled by an active galactic nucleus (AGN) powered by a supermassive black hole. We have accurately determined the local radio luminosity function (RLF) at 1.4GHz for both classes of radio source and have found it to agree well with other recent determinations. From the RLF of star-forming galaxies we derive a local star formation density of 0.022+/-0.001M_{sun}_/yr/Mpc^3^, in broad agreement with recent determinations at radio and other wavelengths.