  We study the nature of the faint radio source population detected in the
MeerKAT International GHz Tiered Extragalactic Exploration (MIGHTEE) Early
Science data in the COSMOS field, focusing on the properties of the radio-loud
active galactic nuclei (AGN). Using the extensive multi-wavelength data
available in the field, we are able to classify 88 per cent of the 5223 radio
sources in the field with host galaxy identifications as AGN (35 per cent) or
star-forming galaxies (54 per cent). We select a sample of radio-loud AGN with
redshifts out to $z \sim 6$ and radio luminosities $10^{20} <
\textrm{L}_{1.4~\textrm{GHz}} / \textrm{W Hz}^{-1} < 10^{27}$ and classify them
as high-excitation and low-excitation radio galaxies (HERGs and LERGs). The
classification catalogue is released with this work. We find no significant
difference in the host galaxy properties of the HERGs and LERGs in our sample.
In contrast to previous work, we find that the HERGs and LERGs have very
similar Eddington-scaled accretion rates; in particular we identify a
population of very slowly accreting AGN that are formally classified as HERGs
at these low radio luminosities, where separating into HERGs and LERGs possibly
becomes redundant. We investigate how black hole mass affects jet power, and
find that a black hole mass $\gtrsim 10^{7.8}~\textrm{M}_\odot$ is required to
power a jet with mechanical power greater than the radiative luminosity of the
AGN ($L_\textrm{mech}/L_\textrm{bol} > 1$). We discuss that both a high black
hole mass and black hole spin may be necessary to launch and sustain a dominant
radio jet.
