pgf/tikz: Making the filling / gap between double border lines transparent

Two more options where you don't need to worry about background color.

The first one (doubleA in code below) uses a matrix node. A matrix node is a node with some other nodes inside. Therefore, if matrix node is only drawn and not filled, it keeps its transparency. Be careful with this option because your internal node must finish with \\.

The second ont (doubleB) uses append after command option to draw a surrounding node with fitting library. Like with doubleA this outer node has no background if you dont fill it.

You can select double line separation adjusting inner sep for matrix or fit nodes. If you need to make further reference to this nodes you will need to select between outer or inner one. With matrix solution you fix outer node name and inner one will be name-1-1, while with fit solution, you fix inner node name and outer one is called name-b (you can change this name).

\documentclass[tikz,border=3mm]{standalone}

\usetikzlibrary{backgrounds,fit,matrix}
\begin{document}

\tikzset{doubleA/.style = {matrix of nodes,
    draw, inner sep=1mm,
    nodes = {rectangle, draw, inner sep=.3333em}}}

\tikzset{doubleB/.style = {rectangle, draw, 
    append after command={
        \pgfextra{\node[fit=(\tikzlastnode), draw] (\tikzlastnode-b) {};}
    }}}

\begin{tikzpicture}

\node[doubleA] at (0,0) (A) {Double\\};

\node[doubleB] at (0,-1)  (B) {Double};

\begin{pgfonlayer}{background}
   \node [fill=black!30,fit={(A) (B-b)}] {};
\end{pgfonlayer}
\end{tikzpicture}

\end{document} 

Update

Another example which shows how to place or connect this kind of nodes. Every node is double, it's made with inner and outer nodes, therefore you can connect inner or outer nodes with any other one. You can also use inner or outer nodes to anchor them.

\documentclass[tikz,border=3mm]{standalone}
\usetikzlibrary{backgrounds,fit,matrix,positioning}
\begin{document}

\tikzset{doubleA/.style = {matrix of nodes,
    draw, inner sep=1mm,
    nodes = {rectangle, draw, inner sep=.3333em}}}

\tikzset{doubleB/.style = {rectangle, draw, 
    append after command={
        \pgfextra{\node[fit=(\tikzlastnode), draw] (\tikzlastnode-b) {};}
    }}}

\begin{tikzpicture}

\fill[red!20] (-1,-1) rectangle (4,4);
\draw[help lines] (-1,-1) grid (4,4);

\node[doubleA] (A)  at (0,0) {DoubleA\\};

\node[doubleB, anchor=south west] at (2,1)  (B) {DoubleB};

\node[doubleA, matrix anchor=south west] at (1,3)  (C) {DoubleA\\};

\node[doubleA, matrix anchor=D-1-1.south west, above=2cm of A.center] (D) {DoubleA\\};

\draw [<->] (A) to [out=80, in=260] (D);
\draw [<->] (D-1-1.north) |-(C-1-1);
\draw [<->] (A-1-1) to [out=10, in=180] (B-b);
\draw [<->] (A) to [out=-10, in=-80] (B);
\end{tikzpicture}
\end{document} 

Easiest is to give the same color to double as that of the background:

\documentclass{article}
\usepackage{tikz}
\usetikzlibrary{backgrounds,fit}
\begin{document}

\begin{tikzpicture}
\node (myNode1) at (0,0) [
                          rectangle,
                          draw,
                          double,
                          double distance=1mm
                        ] {Double};
%
\node (myNode2) at (0,-10mm) [
                          rectangle,
                          draw,
                          double=black!30,                          
                          double distance=1mm,
                          fill=green!30
                        ] {Double};
%
\begin{pgfonlayer}{background}
   \node [fill=black!30,fit=(myNode1) (myNode2)] {};
\end{pgfonlayer}
\end{tikzpicture}

\end{document} 

If you aren't happy with this, then you have to hack into the inner code as there is no relevant key involved. You may also define a new command.