  This paper presents a study of the local radio source population, by
cross-comparing the data from the first data release (DR1) of the LOFAR
Two-Metre Sky Survey (LoTSS) with the Sloan Digital Sky Survey (SDSS) DR7 main
galaxy spectroscopic sample. The LoTSS DR1 provides deep data (median rms noise
of 71 $\mathrm{\mu}$Jy at 150 MHz) over 424 square degrees of sky, which is
sufficient to detect 10615 (32 per cent) of the SDSS galaxies over this sky
area. An improved method to separate active galactic nuclei (AGN) accurately
from sources with radio emission powered by star formation (SF) is developed
and applied, leading to a sample of 2121 local ($z < 0.3$) radio AGN. The local
150 MHz luminosity function is derived for radio AGN and SF galaxies
separately, and the good agreement with previous studies at 1.4 GHz suggests
that the separation method presented is robust. The prevalence of radio AGN
activity is confirmed to show a strong dependence on both stellar and black
hole masses, remarkably reaching a fraction of 100 per cent of the most massive
galaxies ($> 10^{11} \mathrm{M_{\odot}}$) displaying radio-AGN activity with
$L_{\rm 150 MHz} \geq 10^{21}$W Hz$^{-1}$; thus, the most massive galaxies are
always switched on at some level. The results allow the full Eddington-scaled
accretion rate distribution (a proxy for the duty cycle) to be probed for
massive galaxies. More than 50 per cent of the energy is released during the
$\le 2$ per cent of the time spent at the highest accretion rates,
$L_{\mathrm{mech}}/L_{\mathrm{Edd}} > 10^{-2.5}$. Stellar mass is shown to be a
more important driver of radio-AGN activity than black hole mass, suggesting a
possible connection between the fuelling gas and the surrounding halo. This
result is in line with models in which these radio AGN are essential for
maintaining the quenched state of galaxies at the centres of hot gas haloes.
