  G\"odel algebras are the Heyting algebras satisfying the axiom $(x \to y)
\vee (y \to x)=1$. We utilize Priestley and Esakia dualities to dually describe
free G\"odel algebras and coproducts of G\"odel algebras. In particular, we
realize the Esakia space dual to a G\"odel algebra free over a distributive
lattice as the, suitably topologized and ordered, collection of all nonempty
closed chains of the Priestley dual of the lattice. This provides a tangible
dual description of free G\"odel algebras without any restriction on the number
of free generators, which generalizes known results for the finitely generated
case. A similar approach allows us to characterize the Esakia spaces dual to
coproducts of arbitrary families of G\"odel algebras. We also establish
analogous dual descriptions of free algebras and coproducts in every variety of
G\"odel algebras. As consequences of these results, we obtain a formula to
compute the depth of coproducts of G\"odel algebras and show that all free
G\"odel algebras are bi-Heyting algebras.
