Displaying selected bibliographic items in the body of the text

You may use biblatex and its \fullcite macro. Note that is not necessary to typeset the full bibliography.

\documentclass{beamer}

\usepackage{biblatex}

\usepackage{filecontents}

\begin{filecontents}{\jobname.bib}
@article{Bli74,
  author = {Blinder, Alan S.},
  year = {1974},
  title = {The economics of brushing teeth},
  journaltitle = {Journal of Political Economy},
  volume = {82},
  number = {4},
  pages = {887--891},
}
\end{filecontents}

\addbibresource{\jobname.bib}

\begin{document}

\begin{frame}
\fullcite{Bli74}
\end{frame}

% \begin{frame}
% \printbibliography
% \end{frame}

\end{document}

The biblatex-mla package doesn't define a \fullcite command (it would have been very helpful to have mentioned that you were using this package in your question). Adapting some code (and naming convention) from the biblatex-apa package, based on Hanging references using \fullcite, here is a solution to your problem. Since MLA uses different author name ordering in footnote references and bibliography references but you want to be able to display both, we define a \fullcitebib command which will insert a properly formatted full bibliography entry in the text and a \fullcitefoot command which will insert a properly formatted footnote-style entry.

\documentclass{beamer}

\usepackage[style=mla,autocite=footnote]{biblatex}
\DeclareAutoCiteCommand{footnote}[f]{\footcite}{\footcites}
\setlength{\bibhang}{1.5em}
\makeatletter
\DeclareCiteCommand{\fullcitefoot}
  {\renewcommand{\finalnamedelim}
   {\ifnum\value{liststop}>2 \finalandcomma\fi\addspace\&\space}%
   \list{}
   {\setlength{\leftmargin}{\bibhang}%
     \setlength{\itemindent}{-\leftmargin}%
     \setlength{\itemsep}{\bibitemsep}%
     \setlength{\parsep}{\bibparsep}}\item}
  {\usedriver
    {\DeclareNameAlias{sortname}{default}}
    {\thefield{entrytype}}\finentry}
  {\item}
  {\endlist\global\undef\bbx@lasthash}

\DeclareCiteCommand{\fullcitebib}
  {\renewcommand{\finalnamedelim}
   {\ifnum\value{liststop}>2 \finalandcomma\fi\addspace\&\space}%
   \list{}
   {\setlength{\leftmargin}{\bibhang}%
     \setlength{\itemindent}{-\leftmargin}%
     \setlength{\itemsep}{\bibitemsep}%
     \setlength{\parsep}{\bibparsep}}\item}
  {\usedriver
    {}
    {\thefield{entrytype}}\finentry}
  {\item}
  {\endlist\global\undef\bbx@lasthash}
\makeatother
\usepackage{filecontents}

\begin{filecontents}{\jobname.bib}
@article{Bli74,
  author = {Blinder, Alan S.},
  year = {1974},
  title = {The economics of brushing teeth},
  journaltitle = {Journal of Political Economy},
  volume = {82},
  number = {4},
  pages = {887--891},
}
\end{filecontents}

\addbibresource{\jobname.bib}

%\renewcommand{\footnotesize}{\scriptsize} uncomment this if you want footnotes smaller
\begin{document}

\begin{frame}
This is a full citation in  footnote style: \fullcitefoot{Bli74}.

This is a full citation in bibliography style: \fullcitebib{Bli74}
\end{frame}

 \begin{frame}
 \printbibliography
 \end{frame}

\end{document}