  The eccentricity of a sub-stellar companion is an important tracer of its
formation history. Directly imaged companions often present poorly constrained
eccentricities. A recently developed prior framework for orbit fitting called
''observable-based priors'' has the advantage of improving biases in derived
orbit parameters for objects with minimal phase coverage, which is the case for
the majority of directly imaged companions. We use observable-based priors to
fit the orbits of 21 exoplanets and brown dwarfs in an effort to obtain the
eccentricity distributions with minimized biases. We present the objects'
individual posteriors compared to their previously derived distributions,
showing in many cases a shift toward lower eccentricities. We analyze the
companions' eccentricity distribution at a population level, and compare this
to the distributions obtained with the traditional uniform priors. We fit a
Beta distribution to our posteriors using observable-based priors, obtaining
shape parameters $\alpha = 1.09^{+0.30}_{-0.22}$ and $\beta =
1.42^{+0.33}_{-0.25}$. This represents an approximately flat distribution of
eccentricities. The derived $\alpha$ and $\beta$ parameters are consistent with
the values obtained using uniform priors, though uniform priors lead to a tail
at high eccentricities. We find that separating the population into high and
low mass companions yields different distributions depending on the
classification of intermediate mass objects. We also determine via simulation
that the minimal orbit coverage needed to give meaningful posteriors under the
assumptions made for directly imaged planets is $\approx$ 15% of the inferred
period of the orbit.
