  We have identified 7824 radio sources from the 1.4GHz NRAO VLA Sky Survey
(NVSS) with galaxies brighter than K=12.75 mag. in the Second Incremental Data
Release of the 6dF Galaxy Survey (6dFGS DR2). The resulting sample of redshifts
and optical spectra for radio sources over an effective sky area of 7076 square
degrees (about 17% 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=0.043. Through visual examination
of 6dF spectra we have identified the dominant mechanism for radio emission
from each galaxy. 60% are fuelled by star-formation and 40% are fuelled by an
active galactic nucleus powered by a supermassive black hole. We have
accurately determined the local radio luminosity function at 1.4GHz for both
classes of radio source and have found it to agree well with other recent
determinations. From the radio luminosity function of star-forming galaxies we
derive a local star formation density of 0.022+/-0.001 M_sun/yr/Mpc^3, in broad
agreement with recent determinations at radio and other wavelengths.
  We have split the radio luminosity function of radio-loud AGN into bins of
absolute K-band magnitude (M_K) and compared this with the underlying K-band
galaxy luminosity function of all 6dFGS galaxies to determine the bivariate
radio-$K$-band luminosity function. We verify that radio-loud AGN
preferentially inhabit the brightest and hence most massive host galaxies and
show that the fraction of all galaxies which host a radio-loud AGN scales as
L_K^2.1, indicative of a similarly strong scaling with black hole mass and
stellar mass.
