biblatex filter on arbitrary field

Note that annote is an alias for annotation, so the answer uses annotation instead.

You can create bibchecks with almost arbitrary logic. See §3.7.9 Bibliography Filters and Checks of the biblatex manual. A bibcheck will contain the special directive \skipentry to discard a particular entry, all entries that do not get to \skipentry will be displayed. (So in a way you don't explicitly tell biblatex which entries you want, you mark those that you don't want.)

With \iffieldequalstr you can check for field contents. Putting that together you can use

\defbibcheck{annotebar}{\iffieldequalstr{annotation}{bar}{}{\skipentry}}
\defbibcheck{annotefoo}{\iffieldequalstr{annotation}{foo}{}{\skipentry}}

to obtain only those entries with an annotation equal to bar and foo, respectively.

\documentclass[british]{article}
\usepackage[T1]{fontenc}
\usepackage[utf8]{inputenc}
\usepackage{babel}
\usepackage{csquotes}
\usepackage[bibencoding=auto,backend=biber,babel=other]{biblatex}

\usepackage{filecontents}
\begin{filecontents*}{\jobname.bib}
@book{Bara2006,
  address = {New York},
  author = {Bara, Judith},
  publisher = {Routledge},
  title = {English Citation entry},
  year = {2006},
  annote = {foo},
}
@book{Bara2007,
  address = {New York},
  author = {Bara, Judith},
  publisher = {Routledge},
  title = {Another English Citation entry},
  year = {2007},
  annote = {bar},
}
\end{filecontents*}

\addbibresource{\jobname.bib}

\defbibcheck{annotebar}{\iffieldequalstr{annotation}{bar}{}{\skipentry}}
\defbibcheck{annotefoo}{\iffieldequalstr{annotation}{foo}{}{\skipentry}}

\begin{document}
\parencite{Bara2006,Bara2007}
\printbibliography[check=annotebar]
\printbibliography[check=annotefoo]
\end{document}

Here is a solution using filters and categories from biblatex:

\documentclass{article}

\usepackage{filecontents}

\begin{filecontents*}{\jobname.bib}
@article{aaa,
    Author = {Aaa, John and Other, An},
    Title = {Here be dragons},
    Journal = {Whatever journal},
    Pages = {123-234},
    Volume = {1},
    Year = {2000},
    Annote = {Journal Papers}
}
@article{bbb,
    Author = {Bbb, Alexandra},
    Title = {Dunjeon \& Dragons},
    Journal = {A journal},
    Pages = {12-24},
    Volume = {1},
    Year = {2001},
    Annote = {Other Papers}
}
@article{ccc,
    Author = {Ccc, Jules},
    Title = {A small title},
    Journal = {A book},
    Pages = {1023-1026},
    Volume = {2},
    Year = {2016},
    Annote = {Books}
}
\end{filecontents*}

\usepackage{biblatex}
\addbibresource{\jobname.bib}

\DeclareBibliographyCategory{Books}
\DeclareBibliographyCategory{JournalPapers}
\DeclareBibliographyCategory{OtherPapers}
\DeclareIndexFieldFormat{Books}{%
  \ifstrequal{Books}{#1}
  {\addtocategory{Books}{\thefield{entrykey}}}
  {
    \ifstrequal{Journal Papers}{#1}
    {\addtocategory{JournalPapers}{\thefield{entrykey}}}
    {
      \ifstrequal{Other Papers}{#1}
      {\addtocategory{OtherPapers}{\thefield{entrykey}}}
      {}
    }
  }
}
\AtDataInput{\indexfield[Books]{annotation}}

\begin{document}
\nocite{*}
\printbibliography[category=Books,title={Books}]
\printbibliography[category=JournalPapers,title={Journal Papers}]
\printbibliography[category=OtherPapers,title={Other Papers}]
\end{document}