How to plot paired smooth histogram/distribution plots?

Here is something using a custom ChartElementFunction

Module[{c = 0},
 half[{{xmin_, xmax_}, {ymin_, ymax_}}, data_, metadata_] := (c++;   
   Map[Reverse[({0, Mean[{xmin, xmax}]} + # {1, (-1)^c})] &, 
    First@Cases[
      First@Cases[InputForm[SmoothHistogram[data, Filling -> Axis]], 
        gc_GraphicsComplex :> Normal[gc], ∞], 
      p_Polygon, ∞], {2}])]

(thanks to @halirutan for reminding me about how to do closures in WL).

data = RandomVariate[NormalDistribution[0, 1], {4, 2, 100}];

DistributionChart[data, BarSpacing -> -1, ChartElementFunction -> half]

Update: Using GeometricTransformations to post-process SmoothHistogram outputs:

ClearAll[halfSH, pairedSH]
halfSH[side : (Left | Right) : Right][data_, o : OptionsPattern[]] := 
 Module[{i = 1, tr = If[side === Left, ReflectionTransform[{-1, 0}], Identity], 
   col = If[side === Left, Blue, Red]}, 
  Graphics[GeometricTransformation[SmoothHistogram[#, Automatic, "PDF", o, 
   Filling -> Axis, FillingStyle -> Lighter@col, PlotStyle -> col][[1]], 
      Composition[TranslationTransform[{i++, 0}], tr, ReflectionTransform[{1, -1}]]], 
     FilterRules[{o}, Options[Graphics]]] & /@ data]

halfSH[side : (Left | Right) : Right][data_, bwkernel__, 
  o : OptionsPattern[]] := 
 Module[{i = 1, tr = If[side === Left, ReflectionTransform[{-1, 0}], Identity], 
   col = If[side === Left, Blue, Red]}, 
  Graphics[GeometricTransformation[SmoothHistogram[#, bwkernel, o, Filling -> Axis, 
   FillingStyle -> Lighter@col, PlotStyle -> col][[1]], 
     Composition[TranslationTransform[{i++, 0}], tr, ReflectionTransform[{1, -1}]]], 
     FilterRules[{o}, Options[Graphics]]] & /@ data]

pairedSH[bw_: Automatic, df_: "PDF"][{d1_, o1 : OptionsPattern[]}, 
 {d2_, o2 : OptionsPattern[]},  o : OptionsPattern[]] := 
 Show[halfSH[Left][d1, bw, df, o1], halfSH[][d2, bw, df, o2], 
  PlotRange -> {{0, 1 + Length@d1}, Automatic}, o, Frame -> True, 
  FrameTicks -> {{Automatic, Automatic}, {Range[Length@d1], 
     Automatic}}, AspectRatio -> 1/GoldenRatio]

Examples:

{data1, data2} = RandomVariate[NormalDistribution[#, #], {4, 1000}] & /@ {2, 1};

pairedSH[][{data1}, {data2}]

pairedSH[{"Adaptive", 0.3, .5}][{data1, FillingStyle->Lighter[Cyan], PlotStyle->Green}, 
 {data2, FillingStyle -> Lighter@Orange, PlotStyle -> Red}]

Original post:

{data1, data2} = RandomVariate[NormalDistribution[#, #], {4, 1000}] & /@ Range[2];

cedf1 = ChartElementDataFunction["SmoothDensity", "Shape" -> "SingleSided"];
cedf2 = ChartElementDataFunction["SmoothDensity", "Shape" -> "FlippedSingleSided"];

Show[DistributionChart[data1, ChartStyle -> Yellow, BarSpacing -> 2, 
      ChartElementFunction -> cedf1, ChartLabels -> {"a", "b", "c", "d"}], 
 DistributionChart[data2, ChartStyle -> Red, BarSpacing -> 2, 
  ChartElementFunction -> cedf2]]

I just followed your approach but rather created tables of the density and associated x-values. I added a shift parameter to violin to allow the placement of each pair of probability density estimates.

violin[data1_, data2_, shift_] := 
 Module[{d1 = SmoothKernelDistribution[data1],
   d2 = SmoothKernelDistribution[data2], x, xrange},
  {xmin1, xmax1} = MinMax[data1];
  {xmin2, xmax2} = MinMax[data2];
  xrange1 = xmax1 - xmin1;
  xrange2 = xmax2 - xmin2;
  (* Create a table of the density values along with the associated x value *)
  pdf1 = Table[{-PDF[d1, x] + shift, x}, {x, xmin1 - 0.2 xrange1, 
     xmax1 + 0.2 xrange1, 1.4 xrange1/100}];
  pdf2 = Table[{PDF[d2, x] + shift, x}, {x, xmin2 - 0.2 xrange2, 
     xmax2 + 0.2 xrange2, 1.4 xrange2/100}];
  (* Construct violin graphic *)
  Show[Graphics[{Darker[Green], EdgeForm[Darker[Green]], 
     Polygon[pdf1]}],
   Graphics[{Orange, EdgeForm[Orange], Polygon[pdf2]}]]]

(* Generate some data *)
data11 = RandomVariate[NormalDistribution[], 100];
data12 = RandomVariate[NormalDistribution[0.5, 1], 100];
data21 = RandomVariate[NormalDistribution[1, 2], 100];
data22 = RandomVariate[NormalDistribution[0.5, 1.5], 100];

Show[ListPlot[{{-1, 3}}, AxesOrigin -> {-1, -8},
  Ticks -> {{{0, "A"}, {2, B}}, Automatic},
  PlotRange -> {{-1, 3}, {-8, 10}}, PlotStyle -> White],
 violin[data11, data12, 0],
 violin[data21, data22, 2],
 ImageSize -> Large]