Second Version
A version via a simple functional shading (no random values...)
Extract from pgfmanual: Warning: These shadings are the least portable of all and they put the heaviest burden of the renderer. They are slow and, possibly, will not print correctly!
The picture below is a screenshot from AdobeReader (ImageMagick, evince, xpdf, okular produce bad results).

\documentclass[tikz]{standalone}
\usetikzlibrary{fadings,patterns,fit}
% use stamp.png to make a fading (a mask)
\begin{tikzfadingfrompicture}[name=stamp]
\node [fill=transparent!0,inner sep=0]
{\includegraphics[width=50pt,height=50pt]{stamp.png}};
\begin{scope}[yshift=-90pt,transform canvas={scale=.2}]
\node[color=gray!50!black,font=\ttfamily] {\today};
\end{scope}
\end{tikzfadingfrompicture}
\colorlet{mycolor}{blue}
\pgfdeclarefunctionalshading{stampfunctional}
{\pgfpointorigin}{\pgfpoint{4cm}{4cm}}{\pgfshadecolortorgb{mycolor}{\myrgb}}{
20 mul
sin 1 add 0.5 mul
exch
20 mul
cos 1 add 0.5 mul
add 0.5 mul
%1 exch sub
dup \myrgbred
1 exch sub mul 1 exch sub exch
dup \myrgbgreen
1 exch sub mul 1 exch sub exch
\myrgbblue
1 exch sub mul 1 exch sub
}
\begin{document}
\begin{tikzpicture}[inner sep=0]
\colorlet{mycolor}{blue!70!black}
% fill a region stampshading
\node[
shading=stampfunctional,
shading angle=-45,
minimum width=4cm,minimum height=4cm]{};
% fade the same region using stamp fading as mask
\node[fill=white,minimum width=4cm,minimum height=4cm,path fading=stamp]{};
\colorlet{mycolor}{lime!70!black}
% fill a region stampshading
\node[
shading=stampfunctional,
shading angle=-45,
minimum width=4cm,minimum height=4cm] at (0,4){};
% fade the same region using stamp fading as mask
\node[fill=white,minimum width=4cm,minimum height=4cm,path fading=stamp] at (0,4){};
\end{tikzpicture}
\end{document}
First version
The blur effect can't be made via PDF. But with fading and shading, you can simulate some imperfections.
Here is an example using stamp.png (your image in black and transparent color):

\documentclass[tikz]{standalone}
\usetikzlibrary{fadings,patterns,fit}
% use stamp.png to make a fading (a mask)
\begin{tikzfadingfrompicture}[name=stamp]
\node [fill=transparent!0,inner sep=0]
{\includegraphics[width=50pt,height=50pt]{stamp.png}};
\begin{scope}[yshift=-90pt,transform canvas={scale=.2}]
\node[color=gray!50!black,font=\ttfamily] {\today};
\end{scope}
\end{tikzfadingfrompicture}
\pgfdeclarehorizontalshading[mycolor]{stampshading}
{4cm}{
color(0cm)=(mycolor);
color(1cm)=(mycolor);
color(1.35cm)=(mycolor);
color(1.40cm)=(white!80!mycolor);
color(1.45cm)=(mycolor);
color(1.85cm)=(mycolor);
color(2.00cm)=(white!80!mycolor);
color(2.05cm)=(mycolor);
color(2.25cm)=(mycolor);
color(2.30cm)=(white!80!mycolor);
color(2.35cm)=(mycolor);
color(2.45cm)=(mycolor);
color(2.50cm)=(white!80!mycolor);
color(2.55cm)=(mycolor);
color(4cm)=(mycolor)
}
\begin{document}
\begin{tikzpicture}[inner sep=0]
\colorlet{mycolor}{blue!70!black}
% fill a region stampshading
\node[
shading=stampshading,
shading angle=-45,
minimum width=4cm,minimum height=4cm]{};
% fade the same region using stamp fading as mask
\node[fill=white,minimum width=4cm,minimum height=4cm,path fading=stamp]{};
\colorlet{mycolor}{lime!70!black}
% fill a region stampshading
\node[
shading=stampshading,
shading angle=-45,
minimum width=4cm,minimum height=4cm] at (0,4){};
% fade the same region using stamp fading as mask
\node[fill=white,minimum width=4cm,minimum height=4cm,path fading=stamp] at (0,4){};
\end{tikzpicture}
\end{document}
The shading is a simple horizontal shading. With a functional shading, the imperfections could be more realist.
Here the stamp.png image (but without transparent color):

coffee4package. – yo' Jan 31 '13 at 00:20But have multiple overlapping TikZ pictures seems excessive. Maybe it is possible similate with a smooth shadow like those of the 'pst-blur' package or another more simpler way.
– Fran Jan 31 '13 at 18:10\today, but leave the transient/smudging up to ImageMagick, which is the perfect command tool for this job. Don't obsess over a single tool, bur rather chain tools withgnu make. That is the *nix way! – Serge Stroobandt Oct 29 '16 at 22:59