How can I hatch the areas delimited by two intersecting Gaussian distribution curves?

I am not sure if these are the regions you are trying to fill, so I will fill all of them and then you can choose which ones do you need:

\documentclass[8pt]{beamer}
\usepackage{tikz}
\usetikzlibrary{patterns}
\usepackage{pgfplots}

\pgfmathdeclarefunction{dnorm}{2}{%
  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}

\begin{document}

\begin{frame}
\begin{tikzpicture}

\begin{axis}[
  domain=-1:12,
  samples=101,
  height=8cm,
  width=12cm, 
  xticklabel=\empty,
  yticklabel=\empty]

\addplot[
  fill=yellow,
  draw=none,
  domain=-1:6,
  stack plots=y
  ] 
  {dnorm(3,1.5)} \closedcycle;

\addplot [
    fill=orange,
    draw=none,
    domain=-1:6,
    stack plots=y
] {min( dnorm(6,1.5) - dnorm(3,1.5),0)} \closedcycle;

\addplot [
    fill=cyan,
    draw=none,
    domain=-1:6,
    stack plots=y
] {max( dnorm(6,1.5) - dnorm(3,1.5),0)} \closedcycle;

\addplot [
    fill=magenta,
    draw=none,
    domain=6:12,
] {max( dnorm(6,1.5),dnorm(3,1.5))} \closedcycle;

\addplot [
    fill=olive,
    draw=none,
    domain=6:12,
] {min( dnorm(6,1.5),dnorm(3,1.5))} \closedcycle;

% Draw curves
\addplot [thin, smooth, color=black] {dnorm(3,1.5)};
\addplot [thin, smooth, color=black] {dnorm(6,1.5)};

% Draw vertical bar:
\draw [red, thick] ({rel axis cs:0,0}-|{axis cs:6,0}) -- ({rel axis cs:0,1}-|{axis cs:6,0});
\end{axis}

\end{tikzpicture}
\end{frame}

\end{document}

A PSTricks solution using the pst-func package:

\documentclass{article}

\usepackage{pst-func}
\usepackage{xfp}

% function
\def\Gauss[#1,#2]#3{\fpeval{1/(sqrt(2*pi)*#2)*exp(-(#3-#1)^2/(2*#2^2))}}

% settings for PSTricks
\psset{
  xunit = 2,
  yunit = 3
}

\begin{document}

\begin{pspicture}(-0.25,-0.15)(4.8,1.8)
% parameters
\def\mueA{2}
\def\mueB{2.7}
\def\sigma{0.4}
% points on curve
\def\pointA{2.1}
\def\pointB{3.1}
  % drawing
  \psaxes[labels = none, ticks = none]{->}(0,0)(4.8,1.8)
  \uput[270](2.4,0){\small Internal Response (a.u.)}
  \rput(0.3,0.7){\psrotate(0,0){90}{\small Probability}}
  \psGauss[mue = \mueA, sigma = \sigma]{0.5}{3.5}
  \pscustom[fillstyle = hlines, hatchsep = 2pt]{%
    \psGauss[mue = \mueB, sigma = \sigma, hatchsep = 1pt]{2.2}{4.4}
    \psline(2.2,0)(2.2,1)}
  \pscustom[fillstyle = vlines]{%
    \psline(2.2,1)(2.2,0)
    \psGauss[mue = \mueB, sigma = \sigma]{0}{2.2}}
  \psline[linewidth = 1.5pt](2.2,0)(2.2,1.3)
  \uput[90](2.2,1.3){\small criterion respons}
  \psline{->}(\fpeval{\pointA-0.3},\fpeval{\Gauss[\mueB,\sigma]{\pointA}+0.2})(\pointA,\Gauss[\mueB,\sigma]{\pointA})
  \uput[90](  \fpeval{\pointA-0.3},\fpeval{\Gauss[\mueB,\sigma]{\pointA}+0.2}){\footnotesize Miss}
  \psline{->}(\fpeval{\pointB+0.3},\fpeval{\Gauss[\mueB,\sigma]{\pointB}+0.2})(\pointB,\Gauss[\mueB,\sigma]{\pointB})
  \uput[90](  \fpeval{\pointB+0.3},\fpeval{\Gauss[\mueB,\sigma]{\pointB}+0.2}){\footnotesize Hit}
\end{pspicture}

\begin{pspicture}(-0.25,-0.15)(4.8,1.8)
% parameters
\def\mueA{2}
\def\mueB{2.7}
\def\sigma{0.4}
% points on curve
\def\pointA{1.6}
\def\pointB{2.8}
  % drawing
  \psaxes[labels = none, ticks = none]{->}(0,0)(4.8,1.8)
  \uput[270](2.4,0){\small Internal Response (a.u.)}
  \rput(0.3,0.7){\psrotate(0,0){90}{\small Probability}}
  \psGauss[mue = \mueB, sigma = \sigma]{0.5}{4.5}
  \pscustom[fillstyle = hlines, hatchsep = 2pt]{%
    \psGauss[mue = \mueA, sigma = \sigma, hatchsep = 1pt]{2.2}{4.4}
    \psline(2.2,0)(2.2,1)}
  \pscustom[fillstyle = vlines]{%
    \psline(2.2,1)(2.2,0)
    \psGauss[mue = \mueA, sigma = \sigma]{0}{2.2}}
  \psline[linewidth = 1.5pt](2.2,0)(2.2,1.3)
  \uput[90](2.2,1.3){\small criterion respons}
  \psline{->}(\fpeval{\pointA-0.3},\fpeval{\Gauss[\mueA,\sigma]{\pointA}+0.2})(\pointA,\Gauss[\mueA,\sigma]{\pointA})
  \uput[120](\fpeval{\pointA-0.3},\fpeval{\Gauss[\mueA,\sigma]{\pointA}+0.2}){\footnotesize Correct rejection}
  \psline{->}(\fpeval{\pointB+0.3},\fpeval{\Gauss[\mueA,\sigma]{\pointB}+0.2})(\pointB,\Gauss[\mueA,\sigma]{\pointB})
  \uput[90](\fpeval{\pointB+0.3},\fpeval{\Gauss[\mueA,\sigma]{\pointB}+0.2}){\footnotesize False alarm}
\end{pspicture}

\end{document}

Notice that you can easily change the position of the arrows pointing on the curve simply by changing the values of \pointA and \pointB, respectively.

Version 1.10 of pgfplots has been released just recently, and it comes with a new solution for the problem to fill the area between plots.

Note that the old solution is still possible and still valid; this here is merely an update which might simplify the task. In order to keep the knowledge base of this site up-to-date, I present a solution based on the new fillbetween library here:

\documentclass{standalone}

\usepackage{pgfplots}

\usetikzlibrary{patterns}

\pgfplotsset{compat=1.9}

% \usetikzlibrary{}
\usepgfplotslibrary{fillbetween}

\pgfmathdeclarefunction{dnorm}{2}{%
  \pgfmathparse{1/(#2*sqrt(2*pi))*exp(-((x-#1)^2)/(2*#2^2))}%
}

\begin{document}
  \begin{tikzpicture}
    \begin{axis}[domain=-3:12,samples=25,height=8cm,width=12cm]

    % Draw curves
    \addplot [name path=g3,thin, smooth, color=black] {dnorm(3,1.5)};
    \addplot [name path=g6,thin, smooth, color=black] {dnorm(6,1.5)};

    % Draw vertical bar:
    \draw [name path=red,red, thick] ({rel axis cs:0,0}-|{axis cs:6,0}) -- ({rel axis cs:0,1}-|{axis cs:6,0});


    \addplot fill between[of=g3 and g6,
        soft clip={domain=-6:6},
        split,
        every segment no 0/.style={yellow},
        every segment no 1/.style={pattern=grid,pattern color=gray},
    ];

    \path[name path=lower,
        %draw=red,ultra thick,
        intersection segments={
            of=g3 and g6,
            sequence=B0 -- A1
        }
    ];
    \path[name path=axis] (axis cs:-10,0) -- (axis cs:16,0);

    \addplot[pattern=north west lines,
        pattern color=orange]
    fill between[
        of=axis and lower,
        soft clip={domain=-6:6}]
    ;

    \end{axis}
    \end{tikzpicture}
\end{document}

This solution labels the two involved input plots by g3 and g6, respectively. The third \addplot command makes uses of the new \usepgfplotslibrary{fillbetween}: it computes the filled region between g3 and g6. In addition, it clips the result to domain=-6:6 ("soft clip" is quite similar to standard clipping, but tailored for this use-case and with a simplified syntax). Consequently, everything right of x=6 is not part of the filled region. The key split means that the filled region should be split into individual segment, and each segment can receive an individual style. In our case, we assign segment styles for the 0st and 1st segment to be yellow and a pattern, respectively.

The following \path instruction computes (but does not draw) the part below the intersection. It does so by means of the new feature intersection segments which is also part of \usepgfplotslibrary{fillbetween}. In this case, the intersection segments consists of B0 which is the first (0th) segment of the second argument of g3 and g6, in other words: the first intersection segment of g6. It connects this part with A1 which is the second (1st) part of the first argument of of=g3 and g6. This result is not drawn and not filled; it is merely associated with the name lower.

The \path[name path=axis] simply assigns a name a path of the x axis.

Finally, the last \addplot fill between fills the region between the axis and lower and fills it with a pattern. Note that this fill between also has a soft clip path which restricts its region to x=-6:6.