How to add a transparent shadow to a block in beamer?

A possible first approach, redefining \beamerboxesrounded and \endbeamerboxesrounded (as defined in beamerbaseboxes.sty) to use a change of opacity through \pgfsetfillopacity; the lines of code that were included are marked with % NEW; the desired opacity can be changed using the \opacity command (initially set to 0.5):

\documentclass{beamer}
\usepackage{tikz}

\newcommand\opacity{0.5}

\makeatletter
\renewcommand\beamerboxesrounded[2][]{%
  \global\let\beamer@firstlineitemizeunskip=\relax%
  \vbox\bgroup%
  \setkeys{beamerboxes}{upper=block title,lower=block body,width=\textwidth,shadow=false}%
  \setkeys{beamerboxes}{#1}%
  {%
    \usebeamercolor{\bmb@lower}%
    \globalcolorstrue%
    \colorlet{lower.bg}{bg}%
  }%
  {%
    \usebeamercolor{\bmb@upper}%
    \globalcolorstrue%
    \colorlet{upper.bg}{bg}%
  }%
  %
  % Typeset head
  %
  \vskip4bp
  \setbox\bmb@box=\hbox{%
    \begin{minipage}[b]{\bmb@width}%
      \usebeamercolor[fg]{\bmb@upper}%
      #2%
    \end{minipage}}%
  \ifdim\wd\bmb@box=0pt%
    \setbox\bmb@box=\hbox{}%
    \ht\bmb@box=1.5pt%
    \bmb@prevheight=-4.5pt%
  \else%
    \wd\bmb@box=\bmb@width%
    \bmb@temp=\dp\bmb@box%
    \ifdim\bmb@temp<1.5pt%
      \bmb@temp=1.5pt%
    \fi%
    \setbox\bmb@box=\hbox{\raise\bmb@temp\hbox{\box\bmb@box}}%
    \dp\bmb@box=0pt%
    \bmb@prevheight=\ht\bmb@box%
  \fi%
  \bmb@temp=\bmb@width%
  \bmb@dima=\bmb@temp\advance\bmb@dima by2.2bp%
  \bmb@dimb=\bmb@temp\advance\bmb@dimb by4bp%
  \hbox{%
    \begin{pgfpicture}{0bp}{+-\ht\bmb@box}{0bp}{+-\ht\bmb@box}
      \ifdim\wd\bmb@box=0pt%
        \color{lower.bg}%
      \else%
        \color{upper.bg}%
      \fi%
      \pgfpathqmoveto{-4bp}{-1bp}
      \pgfpathqcurveto{-4bp}{1.2bp}{-2.2bp}{3bp}{0bp}{3bp}
      \pgfpathlineto{\pgfpoint{\bmb@temp}{3bp}}
      \pgfpathcurveto%
      {\pgfpoint{\bmb@dima}{3bp}}%
      {\pgfpoint{\bmb@dimb}{1.2bp}}%
      {\pgfpoint{\bmb@dimb}{-1bp}}%
      \bmb@dima=-\ht\bmb@box%
      \advance\bmb@dima by-2pt%
      \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
      \pgfpathlineto{\pgfpoint{-4bp}{\bmb@dima}}
      \pgfusepath{fill}
    \end{pgfpicture}%
    \copy\bmb@box%
  }%
  \nointerlineskip%
  \vskip-1pt%
  \ifdim\wd\bmb@box=0pt%
  \else%
  \hbox{%
    \begin{pgfpicture}{0pt}{0pt}{\bmb@width}{6pt}
      \bmb@dima=\bmb@width%
      \advance\bmb@dima by8bp%
      \pgfpathrectangle{\pgfpoint{-4bp}{-1bp}}{\pgfpoint{\bmb@dima}{8bp}}
      \pgfusepath{clip}
      {\pgftransformshift{\pgfpoint{-4bp}{0bp}}\pgftext[left,base]{\pgfuseshading{bmb@transition}}}%
    \end{pgfpicture}%
  }%
  \nointerlineskip%
  \vskip-0.5pt%
  \fi%
  \pgfsetfillopacity{\opacity}% NEW
  \ifbmb@shadow
    \setbox\bmb@boxshadow=\hbox{\pgfuseshading{bmb@shadow}}%
    \setbox\bmb@boxshadowball=\hbox{\pgfuseshading{bmb@shadowball}}%
    \setbox\bmb@boxshadowballlarge=\hbox{\pgfuseshading{bmb@shadowballlarge}}%
  \fi%
\setbox\bmb@colorbox=\hbox{{\pgfpicturetrue\pgfsetcolor{lower.bg}}}%
  \setbox\bmb@box=\hbox\bgroup\begin{minipage}[b]{\bmb@width}%
    \pgfsetfillopacity{1}% NEW
    \vskip2pt%
    \usebeamercolor[fg]{\bmb@lower}%
    \colorlet{beamerstructure}{upper.bg}%
    \colorlet{structure}{upper.bg}%
    %\color{.}%
  }

\def\endbeamerboxesrounded{%
  \end{minipage}\egroup%
  \wd\bmb@box=\bmb@width%
  \bmb@temp=\dp\bmb@box%
  \advance\bmb@temp by.5pt%
  \setbox\bmb@box=\hbox{\raise\bmb@temp\hbox{\box\bmb@box}}%
  \dp\bmb@box=0pt%
  \bmb@temp=\wd\bmb@box%
  \bmb@dima=\bmb@temp\advance\bmb@dima by2.2bp%
  \bmb@dimb=\bmb@temp\advance\bmb@dimb by4bp%
  \hbox{%
    \begin{pgfpicture}{0bp}{0bp}{0bp}{0bp}
      \ifbmb@shadow%
      {\pgftransformshift{\pgfpoint{4bp}{-3bp}}\pgftext{\copy\bmb@boxshadowball}}
      \begin{pgfscope}
        {%
\pgfsetfillopacity{1}% NEW
        \advance\bmb@temp by-1bp%
        \pgfpathrectangle{\pgfpoint{\bmb@temp}{-7bp}}{\pgfpoint{9bp}{9bp}}%
        \pgfusepath{clip}
        }%
        {\pgftransformshift{\pgfpoint{\bmb@temp}{1bp}}\pgftext{\box\bmb@boxshadowballlarge}}
      \end{pgfscope}
      \begin{pgfscope}
      \pgfsetfillopacity{\opacity}% NEW
        \advance\bmb@temp by-4bp%
        \pgfpathrectangle{\pgfpoint{4bp}{-7bp}}{\pgfpoint{\bmb@temp}{5bp}}
        \pgfusepath{clip}
        {\pgftransformshift{\pgfpoint{4bp}{-7bp}}\pgftext[left,base]{\copy\bmb@boxshadow}}%
      \end{pgfscope}
      \begin{pgfscope}
        \pgfsetfillopacity{\opacity}% NEW
        \advance\bmb@temp by 4bp%
        \bmb@dima=\ht\bmb@box%
        \advance\bmb@dima by\bmb@prevheight%
        \advance\bmb@dima by 4bp%
        \pgfpathrectangle{\pgfpoint{\bmb@temp}{1bp}}{\pgfpoint{4bp}{\bmb@dima}}
        \pgfusepath{clip}
        \advance\bmb@dima by-4bp%
        {\pgftransformshift{\pgfpoint{\bmb@temp}{\bmb@dima}}\pgftext{\box\bmb@boxshadowball}}
        \advance\bmb@dima by-1bp%
        \pgfpathrectangle{\pgfpoint{\bmb@temp}{1bp}}{\pgfpoint{4bp}{\bmb@dima}}
        \pgfusepath{clip}
        \advance\bmb@temp by4bp%
        {\pgftransformshift{\pgfpoint{\bmb@temp}{-3bp}}%
          \pgftransformrotate{90}%
          \pgftext[left,base]{\box\bmb@boxshadow}}%
      \end{pgfscope}
      \fi%
      \pgfsetfillopacity{1}% NEW
      \unhbox\bmb@colorbox%
      \pgfpathqmoveto{-4bp}{1bp}
      \pgfpathqcurveto{-4bp}{-1.2bp}{-2.2bp}{-3bp}{0bp}{-3bp}
      \pgfpathlineto{\pgfpoint{\the\bmb@temp}{-3bp}}
      \pgfpathcurveto%
      {\pgfpoint{\the\bmb@dima}{-3bp}}%
      {\pgfpoint{\the\bmb@dimb}{-1.2bp}}%
      {\pgfpoint{\the\bmb@dimb}{1bp}}%
      {
      \bmb@dima=\ht\bmb@box%
      \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
      \pgfpathlineto{\pgfpoint{-4bp}{\bmb@dima}}
      \pgfusepath{fill}
      }
      \ifbmb@shadow%
      {
        \color{black!50!bg}
        \pgfsetlinewidth{0pt}
        \pgfpathmoveto{\pgfpoint{\bmb@dimb}{-.5bp}}
        \bmb@dima=\ht\bmb@box%
        \advance\bmb@dima by\bmb@prevheight%
        \advance\bmb@dima by 1bp%
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfusepath{stroke}
        \bmb@temp=\bmb@dima
        \advance\bmb@dima by 1bp%
        \color{black!31!bg}
        \pgfpathmoveto{\pgfpoint{\bmb@dimb}{\bmb@temp}}
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfusepath{stroke}
        \advance\bmb@dima by 1bp%
        \advance\bmb@temp by 1bp%
        \color{black!19!bg}
        \pgfpathmoveto{\pgfpoint{\bmb@dimb}{\bmb@temp}}
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfusepath{stroke}
        \advance\bmb@dima by 1bp%
        \advance\bmb@temp by 1bp%
        \color{black!6!bg}
        \pgfpathmoveto{\pgfpoint{\bmb@dimb}{\bmb@temp}}
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfusepath{stroke}
        \advance\bmb@dima by 1.5bp%
        \advance\bmb@temp by 1bp%
        \color{bg}
        \pgfpathmoveto{\pgfpoint{\bmb@dimb}{\bmb@temp}}
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfusepath{stroke}
      }
      \fi
    \end{pgfpicture}%
    \box\bmb@box%
  }%
  \ifbmb@shadow%
    \vskip4bp minus 2bp%
  \else%
    \vskip2bp%
  \fi%
  \egroup% of \vbox\bgroup
}
\makeatother

% background image
\setbeamertemplate{background canvas}{%
  \tikz[remember picture,overlay]%
  \node at (current page) {\includegraphics[height=\paperheight]{example-image}};
}

% blocks with shadow
\setbeamertemplate{blocks}[rounded][shadow=true]

\begin{document}
\begin{frame}
  \frametitle{Block with shadow and with background image}

  \begin{block}{Block with shadow}
    Content...
  \end{block}
\end{frame}
\end{document}

To get real transparency for box shadows with beamer, you could use this patch from the beamer repository. At this time, this patch is not included in beamer (see discussion).

If you use pdflatex, you can use the patched version of beamerbaseboxes.sty.

Here is the result of my MWE:

Here is the patched version of beamerbaseboxes.sty (put it in the same directory as your beamer document to get real transparency):

% Copyright 2003--2007 by Till Tantau
% Copyright 2010 by Vedran Mileti\'c
%
% This file may be distributed and/or modified
%
% 1. under the LaTeX Project Public License and/or
% 2. under the GNU Public License.
%
% See the file doc/licenses/LICENSE for more details.

\def\beamerboxesdeclarecolorscheme#1#2#3{% scheme name, upper color, lower color
  \setbeamercolor{@scheme upper #1}{fg=white,bg={#2}}
  \setbeamercolor{@scheme lower #1}{bg={#3}}
}

\define@key{beamerboxes}{scheme}{\def\bmb@upper{@scheme upper #1}\def\bmb@lower{@scheme lower #1}}
\define@key{beamerboxes}{upper}{\def\bmb@upper{#1}}
\define@key{beamerboxes}{lower}{\def\bmb@lower{#1}}
\define@key{beamerboxes}{width}{\edef\bmb@width{#1}}
\define@key{beamerboxes}{shadow}[true]{\csname bmb@shadow#1\endcsname}

\newif\ifbmb@shadow
\newbox\bmb@box
\newbox\bmb@colorbox
\newdimen\bmb@boxwidth
\newdimen\bmb@boxheight
\newdimen\bmb@prevheight
\newdimen\bmb@temp
\newdimen\bmb@dima
\newdimen\bmb@dimb

\newcommand\beamerboxesrounded[2][]{%
  \global\let\beamer@firstlineitemizeunskip=\relax%
  \vbox\bgroup%
  \setkeys{beamerboxes}{upper=block title,lower=block body,width=\textwidth,shadow=false}%
  \setkeys{beamerboxes}{#1}%
  {%
    \usebeamercolor{\bmb@lower}%
    \globalcolorstrue%
    \colorlet{lower.bg}{bg}%
  }%
  {%
    \usebeamercolor{\bmb@upper}%
    \globalcolorstrue%
    \colorlet{upper.bg}{bg}%
  }%
  %
  % Typeset head
  %
  \vskip4bp
  \setbox\bmb@box=\hbox{%
    \begin{minipage}[b]{\bmb@width}%
      \usebeamercolor[fg]{\bmb@upper}%
      #2%
    \end{minipage}}%
  \ifdim\wd\bmb@box=0pt%
    \setbox\bmb@box=\hbox{}%
    \ht\bmb@box=1.5pt%
    \bmb@prevheight=-4.5pt%
  \else%
    \wd\bmb@box=\bmb@width%
    \bmb@temp=\dp\bmb@box%
    \ifdim\bmb@temp<1.5pt%
      \bmb@temp=1.5pt%
    \fi%
    \setbox\bmb@box=\hbox{\raise\bmb@temp\hbox{\box\bmb@box}}%
    \dp\bmb@box=0pt%
    \bmb@prevheight=\ht\bmb@box%
  \fi%
  \bmb@temp=\bmb@width%
  \bmb@dima=\bmb@temp\advance\bmb@dima by2.2bp%
  \bmb@dimb=\bmb@temp\advance\bmb@dimb by4bp%
  \hbox{%
    \begin{pgfpicture}{0bp}{+-\ht\bmb@box}{0bp}{+-\ht\bmb@box}
      \ifdim\wd\bmb@box=0pt%
        \color{lower.bg}%
      \else%
        \color{upper.bg}%
      \fi%
      \pgfpathqmoveto{-4bp}{-1bp}
      \pgfpathqcurveto{-4bp}{1.2bp}{-2.2bp}{3bp}{0bp}{3bp}
      \pgfpathlineto{\pgfpoint{\bmb@temp}{3bp}}
      \pgfpathcurveto%
      {\pgfpoint{\bmb@dima}{3bp}}%
      {\pgfpoint{\bmb@dimb}{1.2bp}}%
      {\pgfpoint{\bmb@dimb}{-1bp}}%
      \bmb@dima=-\ht\bmb@box%
      \advance\bmb@dima by-2pt%
      \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
      \pgfpathlineto{\pgfpoint{-4bp}{\bmb@dima}}
      \pgfusepath{fill}
    \end{pgfpicture}%
    \copy\bmb@box%
  }%
  \nointerlineskip%
  \vskip-1pt%
  \ifdim\wd\bmb@box=0pt%
  \else%
    \hbox{%
      \begin{pgfpicture}{0pt}{0pt}{\bmb@width}{6pt}
        \bmb@dima=\bmb@width%
        \advance\bmb@dima by8bp%
        \pgfpathrectangle{\pgfpoint{-4bp}{-1bp}}{\pgfpoint{\bmb@dima}{8bp}}
        \pgfusepath{clip}
        {%
          \pgftransformshift{\pgfpoint{-4bp}{0bp}}%
          \pgftext[left,base]{\pgfuseshading{bmb@transition}}%
        }%
      \end{pgfpicture}%
    }%
    \nointerlineskip%
    \vskip-0.5pt%
  \fi%
  \setbox\bmb@colorbox=\hbox{{\pgfpicturetrue\pgfsetcolor{lower.bg}}}%
  \setbox\bmb@box=\hbox\bgroup\begin{minipage}[b]{\bmb@width}%
    \vskip2pt%
    \usebeamercolor[fg]{\bmb@lower}%
    \colorlet{beamerstructure}{upper.bg}%
    \colorlet{structure}{upper.bg}%
    %\color{.}%
  }

\def\endbeamerboxesrounded{%
  \end{minipage}\egroup%
  \bmb@temp=\dp\bmb@box%
  \advance\bmb@temp by.5pt%
  \setbox\bmb@box=\hbox{\raise\bmb@temp\hbox{\box\bmb@box}}%
  \dp\bmb@box=0pt%
  \bmb@boxwidth=\bmb@width%
  \bmb@boxheight=\ht\bmb@box%
  \advance\bmb@boxheight by4bp%
  \advance\bmb@boxheight by\bmb@prevheight%
  \ifbmb@shadow%
    \pgfdeclareradialshading{bmb@shadowball}{\pgfpointorigin}
    {%
      color(0bp)=(pgftransparent!50);
      color(4bp)=(pgftransparent!100)
    }%
    \pgfdeclareradialshading{bmb@shadowballlarge}{\pgfpointorigin}
    {%
      color(0bp)=(pgftransparent!0);
      color(8bp)=(pgftransparent!100)
    }%
    \pgfdeclarehorizontalshading{bmb@shadowhorz}{\bmb@boxheight-6bp}
    {%
      color(0bp)=(pgftransparent!0);
      color(8bp)=(pgftransparent!100)
    }%
    \pgfdeclareverticalshading{bmb@shadowvert}{\bmb@boxwidth-4bp}
    {%
      color(0bp)=(pgftransparent!100);
      color(8bp)=(pgftransparent!0)
    }%
    \pgfdeclarefading{bmb@shadowmask}
    {%
      \begin{pgfpicture}
        \pgftext[at=\pgfpoint{4bp}{4bp}]{\pgfuseshading{bmb@shadowball}}
        \pgftext[at=\pgfpoint{\bmb@boxwidth}{8bp}]{\pgfuseshading{bmb@shadowballlarge}}
        \pgftext[at=\pgfpoint{\bmb@boxwidth+4bp}{\bmb@boxheight+2bp}]{\pgfuseshading{bmb@shadowball}}
        \pgftext[left, at=\pgfpoint{4bp}{4bp}]{\pgfuseshading{bmb@shadowvert}}
        \pgftext[base, at=\pgfpoint{\bmb@boxwidth+4bp}{8bp}]{\pgfuseshading{bmb@shadowhorz}}
        %
        % clipping is needed because shadow is typeset on top of box
        \begin{pgfscope}
          \pgfsetcolor{black}
          \pgfpathrectangle{\pgfpoint{4bp}{8bp}}{\pgfpoint{\[email protected]}{\bmb@boxheight-2bp}}
          \pgfusepath{fill}
        \end{pgfscope}
      \end{pgfpicture}
    }%
  \fi%
  \bmb@temp=\bmb@width%
  \bmb@dima=\bmb@temp\advance\bmb@dima by2.2bp%
  \bmb@dimb=\bmb@temp\advance\bmb@dimb by4bp%
  \hbox{%
    \begin{pgfpicture}{0bp}{0bp}{0bp}{0bp}
      \ifbmb@shadow%
        \begin{pgfscope}
          \pgfpathrectangle{\pgfpoint{0bp}{-7bp}}
            {\pgfpoint{\bmb@boxwidth+8bp}{\bmb@boxheight+6bp}}
          \pgfsetfading{bmb@shadowmask}{%
            \pgftransformshift{\pgfpoint{0.5\bmb@boxwidth+6bp}{0.5\bmb@boxheight-4bp}}}
          \pgfusepath{fill}
        \end{pgfscope}
      \fi%
      \unhbox\bmb@colorbox%
      \pgfpathqmoveto{-4bp}{1bp}
      \pgfpathqcurveto{-4bp}{-1.2bp}{-2.2bp}{-3bp}{0bp}{-3bp}
      \pgfpathlineto{\pgfpoint{\the\bmb@temp}{-3bp}}
      \pgfpathcurveto%
      {\pgfpoint{\the\bmb@dima}{-3bp}}%
      {\pgfpoint{\the\bmb@dimb}{-1.2bp}}%
      {\pgfpoint{\the\bmb@dimb}{1bp}}%
      {
        \bmb@dima=\ht\bmb@box%
        \pgfpathlineto{\pgfpoint{\bmb@dimb}{\bmb@dima}}
        \pgfpathlineto{\pgfpoint{-4bp}{\bmb@dima}}
        \pgfusepath{fill}
      }
    \end{pgfpicture}%
    \box\bmb@box%
  }%
  \ifbmb@shadow%
    \vskip4bp minus 2bp%
  \else%
    \vskip2bp%
  \fi%
  \egroup% of \vbox\bgroup
}

%
% Shadings
%

\pgfdeclareverticalshading[lower.bg,upper.bg]{bmb@transition}{200cm}%
{%
  color(0pt)=(lower.bg);
  color(2pt)=(lower.bg);
  color(4pt)=(upper.bg)
}

This problem is fixed in beamer v. 3.51 (and only this version), see https://github.com/josephwright/beamer/pull/469

Your code now gives the following output:

If you have another beamer version, try to place the following two files in your working directory:

• https://github.com/josephwright/beamer/blob/d1d9094893ef705bc88ed60f64e4f83a1b1a6b85/base/beamerbaseboxes.sty
• https://github.com/josephwright/beamer/blob/d1d9094893ef705bc88ed60f64e4f83a1b1a6b85/base/themes/outer/beamerouterthemeshadow.sty

It won't work for dvi-based compilation chains.