How to create a twocolored checkerboard pattern with tikz?

You can start from already defined checkerboard light grey which uses two colors. It draws a large square and two smaller squares with a second color on it. Based is this definition and adapting it to your code is easy to do

\documentclass[tikz]{standalone}
\usetikzlibrary{patterns}
\usepackage{luatex85}

% Flexible Checkerboards - based on the definition for checkerboard, but with different size.
% Define size of own check in pattern
\newlength{\flexcheckerboardsize}

%Define a new checkerboard with predefined size and colors.
%Parameters: Name, size, color1, color2
\newcommand{\defineflexcheckerboard}[4]{
    \setlength{\flexcheckerboardsize}{#2}
    \pgfdeclarepatterninherentlycolored{#1}
        {\pgfpointorigin}{\pgfqpoint{2\flexcheckerboardsize}    
        {2\flexcheckerboardsize}}
        {\pgfqpoint{2\flexcheckerboardsize}
        {2\flexcheckerboardsize}}%
        {
            \pgfsetfillcolor{#4}
            \pgfpathrectangle{\pgfpointorigin}{
            \pgfqpoint{2.1\flexcheckerboardsize}    
                {2.1\flexcheckerboardsize}}% make
                                % slightly larger to ensure that tiles
                                % are really solid
          \pgfusepath{fill}
          \pgfsetfillcolor{#3}
          \pgfpathrectangle{\pgfpointorigin}
            {\pgfqpoint{\flexcheckerboardsize}
            {\flexcheckerboardsize}}
          \pgfpathrectangle{\pgfqpoint{\flexcheckerboardsize}
            {\flexcheckerboardsize}}
            {\pgfqpoint{\flexcheckerboardsize}
            {\flexcheckerboardsize}}
            \pgfusepath{fill}
        }
}

\defineflexcheckerboard{flexcheckerboard_bw}{.5mm}{black}{white}
\defineflexcheckerboard{flexcheckerboard_redblue}{.5mm}{red}{blue}
\defineflexcheckerboard{flexcheckerboard_greenorange}{1mm}{green}{orange}
\defineflexcheckerboard{flexcheckerboard_bluecyan}{.2mm}{cyan}{blue}

\begin{document}
  \begin{tikzpicture}[node distance=5mm]
  %background is not used
  \node (c0bw) [circle, radius = 1cm, pattern=flexcheckerboard_bw] {};
  \node (c0rb) [circle, radius = 1cm, pattern=flexcheckerboard_redblue,right of=c0bw] {}; %Wrong no blue
  \node (c0go) [circle, radius = 1cm, pattern=flexcheckerboard_greenorange,below of=c0bw] {}; %Wrong no blue
  \node (c0og) [circle, radius = 1cm, pattern=flexcheckerboard_bluecyan,below of=c0rb] {};   \end{tikzpicture}
\end{document}

In Tikz : rotate a fill pattern some other bicolor patterns are defined. They can also be adapted to your syntax.

Just spelling out my comment a bit (and using this answer). TikZ has already a pattern color, which gets stored in \tikz@pattern@color. (To use it, I put the pattern in a \makeatletter environment.) And I do not see anything wrong with using a background, you can add the pattern in a postaction and get beautiful two-colored patterns. However, after I learned that this might potentially be used in the famous tikzducks package, I rewrote my code to come with two functionalities: either draw a two-color checkerboard or just one color on a background (checker board background=none, which is the predefined setting). You can adjust all the things with pgfkeys, just like in ordinary patterns.

\documentclass{standalone}
\usepackage{tikz}
\usetikzlibrary{patterns}

\makeatletter
\tikzset{/tikz/.cd,
    checker board size/.store in=\flexcheckerboardsize,
    checker board size=6pt,
    checker board background/.store in=\flexcheckerboardbgd,
    checker board background=none
}

\pgfdeclarepatterninherentlycolored[\flexcheckerboardsize,\flexcheckerboardbgd]{flex checker board}
{\pgfqpoint{-1pt}{-1pt}}
{\pgfqpoint{\flexcheckerboardsize}{\flexcheckerboardsize}}
{\pgfqpoint{\flexcheckerboardsize}{\flexcheckerboardsize}}
{\pgfmathsetmacro{\tmp}{0.5*\flexcheckerboardsize}
    \pgfsetcolor{\tikz@pattern@color}
    \pgfpathrectangle{\pgfpointorigin}{\pgfqpoint{\tmp pt}{\tmp pt}}
    \pgfpathrectangle{\pgfqpoint{\tmp pt}{\tmp pt}}{\pgfqpoint{\tmp pt}{\tmp pt}}
    \pgfusepath{fill}
    \def\none{none}
    \ifx\flexcheckerboardbgd\none%
    \else
    \pgfsetcolor{\flexcheckerboardbgd}
    \pgfpathrectangle{\pgfqpoint{0pt}{\tmp pt}}{\pgfqpoint{\tmp pt}{\tmp pt}}
    \pgfpathrectangle{\pgfqpoint{\tmp pt}{0pt}}{\pgfqpoint{\tmp pt}{\tmp pt}}
    \pgfusepath{fill}
    \fi
}
\makeatother

\begin{document}
\begin{tikzpicture}

    \draw [checker board size=9pt, pattern=flex checker board, pattern color =
    red] (0,0) circle[radius = 1cm];
    \draw [checker board background=blue,checker board size=12pt, pattern=flex checker board, pattern color =
    red] (2,0) circle[radius = 1cm];
    \draw [fill=orange,postaction={checker board background=none,checker board size=12pt, pattern=flex checker board, pattern color =
    red}] (4,0) circle[radius = 1cm];

\end{tikzpicture}
\end{document}