The use and position of notes depends, partially, on the biblatex style. Here is a solution independent of each style, except that it requires biber as the backend. The solution involves the use of the standard field note (or the custom field mynote) to store the text to be used for each entry's note.
\documentclass{article}
\usepackage{filecontents}
\begin{filecontents}{test.bib}
@article{article01,
Author = {John Doe},
Journal = {Journal name},
Pages = {280--291},
Title = {Title},
Volume = {21},
Number = {1},
Year = {2013},
note = {My contribution to this paper was \dots}}
@article{article02,
Author = {John Doe},
Journal = {Journal name},
Pages = {280--291},
Title = {Title},
Volume = {21},
Number = {1},
Year = {2014},
note = {My contribution to this paper was \dots\dots}}
\end{filecontents}
\begin{filecontents}{biblatex-dm.cfg}
\DeclareDatamodelFields[type=field,datatype=literal,skipout=false]{mynote}
\end{filecontents}
\usepackage[style=chem-rsc]{biblatex}
\addbibresource{test.bib}
\DeclareSourcemap{
\maps[datatype=bibtex]{
\map{
\step[fieldsource=note]
\step[fieldset=mynote, origfieldval]
\step[fieldset=note, null]
}
}
}
\AtEveryBibitem{
\clearfield{note}
\csappto{blx@bbx@\thefield{entrytype}}{\iffieldundef{mynote}{}{\par\printfield{mynote}}}
}
\begin{document}
\nocite{*}
\printbibliography
\end{document}
The block
\begin{filecontents}{biblatex-dm.cfg}
\DeclareDatamodelFields[type=field,datatype=literal,skipout=false]{mynote}
\end{filecontents}
generates the biblatex configuration file bib latex-dm.cfg where we specify that we create a new custom field mynote.
Then we use the ability of biber to manipulate the bib files to copy the value of the note field to the entry mynote. This is achieved by the block
\DeclareSourcemap{
\maps[datatype=bibtex]{
\map{
\step[fieldsource=note]
\step[fieldset=mynote, origfieldval]
\step[fieldset=note, null]
}
}
}
which also empties note.
Then we have to give the instruction of how and where to print the notes.
\AtEveryBibitem{
\clearfield{note}
\csappto{blx@bbx@\thefield{entrytype}}{\iffieldundef{mynote}{}{\par\printfield{mynote}}}
}
For every bibitem, we clear the value of note; thus, one has the option to operate using biber or directly in biblatex. Then we patch the driver for the current entry by appending to it the text of a note (mynote).

biblatexstyle used. Do you usenoteorannotefor annoting the entries in yourbibfiles. Whichbiblatexstyle of you use? – Guido Jan 11 '13 at 16:26chem-rsc. My entries are not annotated so the fieldsnoteandannoteare free to be used for this. – Darling Jan 11 '13 at 16:34notefield as it was the first one I saw in jabref. – Darling Jan 11 '13 at 16:52