documentclass[tikz,border=7pt]{standalone}
usetikzlibrary{decorations.markings,intersections}
tikzset{
  intersection/times/.code={
    defa{.25} % <- mock value
    defb{.75} % <- mock value
  },
  test mark/.style 2 args={ % <- mark two posions on the path
    decoration={ markings,
      mark =at position #1 with {draw[purple,-latex](0,0) -- (1,0);},
      mark =at position #2 with {draw[purple,-latex](0,0) -- (-1,0);}
    }, decorate
  },
  dot/.style={insert path={node[inner sep=1pt,circle,fill=red]{}}}
}
deftestpath{(1,0) arc(270:90:1cm)}
begin{document}
  begin{tikzpicture}[scale=2]
    draw[red, name path=arc] testpath;
    draw[blue, densely dotted, name path=line] (0,0) -- (1,1) --(0,2);
    draw[
        name intersections={
          of=arc and line, sort by=arc,
          by={a,b},
          times={a,b} % <- this is a mock style that recover the intersection times
        },
        test mark={a}{b} % <- mark at the intersection times
      ] testpath (a)[dot] (b)[dot]; % <- mark at the intersection positions
  end{tikzpicture}
end{document}

documentclass[tikz,border=7pt]{standalone}
usetikzlibrary{decorations.markings,intersections}
usepackage{pgfplots} %<-added
usepgfplotslibrary{fillbetween} %<-added
pgfplotsset{compat=1.16} %<-added

tikzset{
  intersection/times/.code={
    defa{.25} % <- mock value
    defb{.75} % <- mock value
  },
  test mark/.style 2 args={ % <- mark two posions on the path
    decoration={ markings,
      mark =at position #1 with {draw[purple,-latex](0,0) -- (1,0);},
      mark =at position #2 with {draw[purple,-latex](0,0) -- (-1,0);}
    }, decorate
  },
  dot/.style={insert path={node[inner sep=1pt,circle,fill=red]{}}}
}
deftestpath{(1,0) arc(270:90:1cm)}
begin{document}
  begin{tikzpicture}[scale=2]
    draw[red, name path=arc] testpath;
    draw[blue, densely dotted, name path=line] (0,0) -- (1,1) --(0,2);
    path[
        name intersections={
          of=arc and line, sort by=arc,
          by={a,b},
          total=Nint
        },
      ]
      pgfextra{typeout{numberspace ofspace intersections:space Nint}
      pgfintersectiongetsolutiontimes{1}{tmp}{dummy}
      pgfmathsetmacro{a}{tmp/2}
      pgfintersectiongetsolutiontimes{2}{tmp}{dummyTwo}
      pgfmathsetmacro{b}{tmp/2}
      typeout{a,b,dummy}}
      testpath (a)[dot] (b)[dot]; % <- mark at the intersection positions

   % desired output with fillbetween library
   path [draw,blue,
    name path=middle arc,
    intersection segments={
        of=arc and line,
        sequence={A1}
    },
    postaction={test mark={0}{1}}];
  end{tikzpicture}
end{document}