  We carried out a Bayesian homogeneous determination of the orbital parameters
of 231 transiting giant planets (TGPs) that are alone or have distant
companions; we employed DE-MCMC methods to analyse radial-velocity (RV) data
from the literature and 782 new high-accuracy RVs obtained with the HARPS-N
spectrograph for 45 systems over 3 years. Our work yields the largest sample of
systems with a transiting giant exoplanet and coherently determined orbital,
planetary, and stellar parameters. We found that the orbital parameters of TGPs
in non-compact planetary systems are clearly shaped by tides raised by their
host stars. Indeed, the most eccentric planets have relatively large orbital
separations and/or high mass ratios, as expected from the equilibrium tide
theory. This feature would be the outcome of high-eccentricity migration (HEM).
The distribution of $\alpha=a/a_R$, where $a$ and $a_R$ are the semi-major axis
and the Roche limit, for well-determined circular orbits peaks at 2.5; this
also agrees with expectations from the HEM. The few planets of our sample with
circular orbits and $\alpha >5$ values may have migrated through disc-planet
interactions instead of HEM. By comparing circularisation times with stellar
ages, we found that hot Jupiters with $a < 0.05$ au have modified tidal quality
factors $10^{5} < Q'_p < 10^{9}$, and that stellar $Q'_s > 10^{6}-10^{7}$ are
required to explain the presence of eccentric planets at the same orbital
distance. As a by-product of our analysis, we detected a non-zero eccentricity
for HAT-P-29; we determined that five planets that were previously regarded to
have hints of non-zero eccentricity have circular orbits or undetermined
eccentricities; we unveiled curvatures caused by distant companions in the RV
time series of HAT-P-2, HAT-P-22, and HAT-P-29; and we revised the planetary
parameters of CoRoT-1b.
