  Let $G$ be a multigraph and $L\,:\,E(G) \to 2^\mathbb{N}$ be a list
assignment on the edges of $G$. Suppose additionally, for every vertex $x$, the
edges incident to $x$ have at least $f(x)$ colors in common. We consider a
variant of local edge-colorings wherein the color received by an edge $e$ must
be contained in $L(e)$. The locality appears in the function $f$, i.e., $f(x)$
is some function of the local structure of $x$ in $G$. Such a notion is a
natural generalization of traditional local edge-coloring. Our main results
include sufficient conditions on the function $f$ to construct such colorings.
As corollaries, we obtain local analogs of Vizing and Shannon's theorems,
recovering a recent result of Conley, Greb\'ik and Pikhurko.
