TikZ clip shapes using several (built in) shapes

No clipping is needed.

\documentclass[tikz]{standalone}

\usetikzlibrary{positioning}
\usetikzlibrary{shapes}

\begin{document}

\begin{tikzpicture}
    \newcommand{\CloudDist}{1.8}

    \tikzstyle {cloudkeys} = [cloud puffs=30, cloud puff arc=150, aspect=1.25, inner sep=0.7cm,] 
    \tikzstyle {mycloud1}   = [draw, cloud, cloudkeys, fill=blue!25]
    \tikzstyle {mycloud2}   = [      cloud, cloudkeys, fill=blue!25]

    \coordinate (cloud 0);
    \coordinate [right=\CloudDist cm of cloud 0] (cloud 1);
    \coordinate [right=\CloudDist cm of cloud 1] (cloud 2);

\begin{scope}[transparency group,nearly opaque]
    \foreach \x in {0,1,2} {
        \node [mycloud1] at (cloud \x) (local map cloud shape \x) {};
    }
    \foreach \x in {0,1,2} {
        \node [mycloud2] at (cloud \x) (local map cloud shape \x) {};
    }
\end{scope}
\end{tikzpicture}
\end{document}

You can use clipping, but the other solution is simpler. Note that the nodes anchors do not form a rectangle.

\documentclass{standalone}

\usepackage{tikz}

\usetikzlibrary{positioning}
\usetikzlibrary{shapes}

\begin{document}

\begin{tikzpicture}
    \newcommand{\CloudDist}{1.8}

    \tikzstyle {cloudkeys} = [cloud puffs=30, cloud puff arc=150, aspect=1.25, inner sep=0.7cm,] 
    \tikzstyle {mycloud}   = [draw, cloud, cloudkeys, fill=blue!25, nearly opaque] 

    \coordinate (cloud 0);
    \coordinate [right=\CloudDist cm of cloud 0] (cloud 1);
    \coordinate [right=\CloudDist cm of cloud 1] (cloud 2);

    \path (cloud 0) -- (cloud 1) coordinate[midway] (cloud edge 1);
    \path (cloud 1) -- (cloud 2) coordinate[midway] (cloud edge 2);

    \begin{scope}
      \node[cloud, cloudkeys,outer sep=2pt] (temp) at (cloud 0) {};% invisible, used for clip rectangle
      \clip (temp.south -| temp.west) rectangle (temp.north -| cloud edge 1);
      \node [mycloud] at (cloud 0) (local map cloud shape 0) {};
    \end{scope}
    \begin{scope}
      \node[cloud, cloudkeys,outer sep=2pt] (temp) at (cloud 1) {};% invisible, used for clip rectangle
      \clip (temp.south -| cloud edge 1) rectangle (temp.north -| cloud edge 2);
      \node [mycloud] at (cloud 1) (local map cloud shape 1) {};
    \end{scope}
    \begin{scope}
      \node[cloud, cloudkeys,outer sep=2pt] (temp) at (cloud 2) {};% invisible, used for clip rectangle
      \clip (temp.south -| cloud edge 2) rectangle (temp.north -| temp.east);
      \node [mycloud] at (cloud 2) (local map cloud shape 2) {};
    \end{scope}

\end{tikzpicture}
\end{document}