  A Schreier decoration is a combinatorial coding of an action of the free
group $F_d$ 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 $\mathbb{Z}^d,d\geq3$, 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.
