Eccentricity of wide binaries is difficult to measure due to their long orbital periods. With Gaia's high-precision astrometric measurements, eccentricity of a wide binary can be constrained by the angle between the separation vector and the relative velocity vector (the v-r angle). In this paper, by using the v-r angles of wide binaries in Gaia Early Data Release 3, we develop a Bayesian approach to measure the eccentricity distribution as a function of binary separations. Furthermore, we infer the eccentricities of individual wide binaries and make them publicly available. Our results show that the eccentricity distribution of wide binaries at 102 AU is close to uniform and becomes superthermal at >103 AU, suggesting two formation mechanisms dominating at different separation regimes. The close binary formation, most likely disc fragmentation, results in a uniform eccentricity distribution at <102 AU. The wide binary formation that leads to highly eccentric wide binaries at >103 AU may be turbulent fragmentation and/or the dynamical unfolding of compact triples. With Gaia, measuring eccentricities is now possible for a large number of wide binaries, opening a new window to understanding binary formation and evolution.