Reduce font size and keep label position in bar chart using PGF/TikZ

I switch my comment to an answer.

You are using every node near coord/.style={font=\tiny},. In this case all predefined settings will be lost. pgfkeys has a special handler to add an extra option to a defined style named append style. Based on this information you can use every node near coord/.append style={font=\tiny}.

\documentclass{article}

\usepackage{pgfplots}
\pgfplotsset{compat=newest}

\begin{document}

\begin{tikzpicture}
\begin{axis}[
    %small,
    ybar,%=8pt, % configures ‘bar shift’
    enlargelimits=0.15,
    ylabel={index},
    symbolic x coords={1982, 1990, 1999, 2006},
    %xtick=data,
    %tick label style={font=\footnotesize},
    legend style={at={(0.5,-0.15)},
    anchor=north,legend columns=-1},
    nodes near coords,
    every node near coord/.append style={font=\tiny},
   nodes near coords align={vertical},
    ]
\addplot coordinates {(1982, 1.78) (1990, 1.71) (1999, 1.68) (2006, 1.62)};
\addplot coordinates {(1982,  1.70) (1990, 1.62) (1999, 1.59) (2006, 1.64)};
\addplot coordinates {(1982, 2.04) (1990, 1.96) (1999, 1.95) (2006, 1.91)};
\legend{USA, Netherlands, {(West-)Germany}}

\end{axis}
\end{tikzpicture}

\end{document}