A Schreier decoration is a combinatorial coding of an action of the free group Fd on the vertex set of a 2d-regular graph. We investigate whether a Schreier decoration exists on various countably infinite transitive graphs as a factor of iid. We show that Zd,d≥3, the square lattice and also the three other Archimedean lattices of even degree have finitary-factor-of-iid Schreier decorations, and exhibit examples of transitive graphs of arbitrary even degree in which obtaining such a decoration as a factor of iid is impossible. We also prove that symmetrical planar lattices with all degrees even have a factor of iid balanced orientation, meaning the indegree of every vertex is equal to its outdegree, and demonstrate that the property of having a factor-of-iid balanced orientation is not invariant under quasi-isometry.