  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 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 non-amenable, quasi-transitive, unimodular $2d$-regular
graphs have a factor of iid balanced orientation, meaning each in- and
outdegree is equal to $d$. This result involves extending earlier spectral
theoretic results on Bernoulli shifts to the Bernoulli graphings of
quasi-transitive, unimodular graphs. Balanced orientation is also obtained for
certain quasi-transitive planar lattices.
