  We present measurements of black hole masses and Eddington ratios for a
sample of 38 bright (M$_{1450}$ < -24.4 mag) quasars at 5.8 < z < 7.5, derived
from VLT/X-shooter near-IR spectroscopy of their broad CIV and MgII emission
lines. The black hole masses (on average M$_{BH}$ ~ 4.6 x 10$^9$ M$_{\odot}$)
and accretion rates (with Eddington ratios ranging between 0.1 and 1.0) are
broadly consistent with that of similarly luminous 0.3 < z < 2.3 quasars, but
there is evidence for a mild increase in the median Eddington ratio going
towards z > 6. Combined with deep ALMA observations of the [CII] 158 $\mu$m
line from the quasar host galaxies and VLT/MUSE investigations of the extended
Ly$\alpha$ halos, this study provides fundamental clues to models of the
formation and growth of the first massive galaxies and black holes. Compared to
local scaling relations, z > 5.7 black holes appear to be over-massive with
respect to their host galaxies, and their accretion properties do not change
with host galaxy morphology. Under the assumption that the kinematics of the T
~ 10$^4$ K gas, traced by the extended Ly$\alpha$ halos, are dominated by the
gravitational potential of the dark matter halo, we report a similar relation
between the black hole mass and circular velocity to the one reported for z ~ 0
galaxies. These results paint a picture where the first supermassive black
holes reside in massive halos at z > 6 and lead the first stages of galaxy
formation by rapidly growing in mass with a duty cycle of order unity. However,
this duty cycle needs to drastically drop towards lower redshifts, while the
host galaxies continue forming stars at a rate of hundreds of solar masses per
year, sustained by the large reservoirs of cool gas surrounding them.
