This uses Alan Munn's suggestion in a comment on the question. I'd tried chapterbib before but got errors, even using the documentation for natbib. However, experimentation suggests the following should work:
\begin{filecontents}{f1.tex}
\chapter{A chapter}
\bibliographystyle{plainnat}
\citestyle{plainnat}
References to an article, \cite{article-full},
and to an entire journal, which appears
only in the references\nocite{whole-journal}.
\bibliography{xampl}
\end{filecontents}
\begin{filecontents}{f2.tex}
\chapter{Another chapter}
\bibliographystyle{plainnat}
\citestyle{plain}
A new chapter with a different style and a cite
of some possibly interesting proceedings, \cite{proceedings-full}.
\bibliography{xampl}
\end{filecontents}
\documentclass[openany]{book}
\usepackage{chapterbib}
\usepackage[sectionbib]{natbib}
\begin{document}
\include{f1}
\include{f2}
\end{document}
Note that various things seem to be crucial here. In particular, you need to issue \citestyle{} which was not obvious to me from the documentation. In particular, you need to do this even when you want to use the plainnat style and even when you've just issued \bibliographystyle{plainnat}. At least, without this, I could not get the correct format to be used.

bibunitscan do this but I've deleted my answer because ti does not allow numerical + author-year ifnatbibis used. – cfr Jan 02 '15 at 03:57chapterbibpackage can do this. See Multiple bibliographies for an example. – Alan Munn Jan 02 '15 at 06:29chapterbibworking in this way before and failed. You should answer, though. – cfr Jan 03 '15 at 00:21