The GaLactic and Extragalactic All-sky Murchison Widefield Array (GLEAM) is a radio continuum survey at 76-227 MHz of the entire southern sky (Declination $<+30deg$) with an angular resolution of $approx 2$ arcmin. In this paper, we combine GLEAM data with optical spectroscopy from the 6dF Galaxy Survey to construct a sample of 1,590 local (median $z approx 0.064$) radio sources with $S_{200,mathrm{MHz}} > 55$ mJy across an area of $approx 16,700~mathrm{deg}^{2}$. From the optical spectra, we identify the dominant physical process responsible for the radio emission from each galaxy: 73 per cent are fuelled by an active galactic nucleus (AGN) and 27 per cent by star formation. We present the local radio luminosity function for AGN and star-forming galaxies at 200 MHz and characterise the typical radio spectra of these two populations between 76 MHz and $sim 1$ GHz. For the AGN, the median spectral index between 200 MHz and $sim 1$ GHz, $alpha_{mathrm{high}}$, is $-0.600 pm 0.010$ (where $S propto u^{alpha}$) and the median spectral index within the GLEAM band, $alpha_{mathrm{low}}$, is $-0.704 pm 0.011$. For the star-forming galaxies, the median value of $alpha_{mathrm{high}}$ is $-0.650 pm 0.010$ and the median value of $alpha_{mathrm{low}}$ is $-0.596 pm 0.015$. Among the AGN population, flat-spectrum sources are more common at lower radio luminosity, suggesting the existence of a significant population of weak radio AGN that remain core-dominated even at low frequencies. However, around 4 per cent of local radio AGN have ultra-steep radio spectra at low frequencies ($alpha_{mathrm{low}} < -1.2$). These ultra-steep-spectrum sources span a wide range in radio luminosity, and further work is needed to clarify their nature.
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