Does the following script match your need?
But there will be more appropriate tweaks to "Start thin, go wide in the middle, end thin again".

// A sample script to change pressure values applied on the foreground path
var pathes = activeDocument.getMatchingItems(Path,{selected: true});
if (pathes.length > 0){
  var p = pathes[0];          // get the foreground path in the selection
  var td = p.getTabletData(); // gettabletdata
 var str; // # (the lines marked '#' were added on 11.02.07)
  if(!td){ // #
    str = "no tabletData";  // #
  } else { // #
    // show the information as a text object
    var str = 'length:' + td.length + "r"; // it's an array.
    for(var i=0; i<td.length; i++){ // Each item has properties.
      str += i + ': offset:' + td[i].offset.toFixed(4) // They are 'offset'
        + '  value:' + td[i].value.toFixed(4) + "r";  // and 'value'.
    }
  } // #
  var ptxt = new PointText(p.bounds.center); // create an text object
  ptxt.range.characterStyle.fontSize = 8;
  ptxt.content = str;

  if(td){ // #
    // now, change the values
    // # it isn't work nicely if there isn't enough number of elements in tabletData
    for(var j=0; j<td.length; j++){
      with(td[j]){
        if(offset < 0.2) value *= offset/0.2;
        else if(offset > 0.8) value *= (1-offset)/0.2;
      }
    }
    
    p.setTabletData(td); // apply it

    // apply same data to the background pathes in the selection #
    if(pathes.length > 1){ // #
      for(var j=1;j<pathes.length;j++){ pathes[j].setTabletData(td); } // #
    } // #
  } // #
}

Praise Scriptographer!!

That looks a lot like what I've been looking for! I shall have to try that out soon! Thanks!

Is there such a script available at the moment?

I tried writing it, by modifying above script.
This script changes the pressure values applied on the selected brushes at both ends.
Usage: Select brush strokes, select this script in Sg's palette, then press Run button on the palette.
Note that minimum thickness of the brush depends on the setting of it.

# last modified: 2007-02-11 : add a quick-fix to ignore strokes that aren't created with a stylus pen.

Thank you, Hiroyuki! I'm impressed you found out about how to use the tablet data feature.
Just a little note: Instead of using the getter / setter functions directly, you can also use the data like this (through the so called bean properties).

var td = path.tabletData;
...
path.tabletData = td;

Also, please note that there is a maximum of supported data. I cannot remember if it was 256, 512 or 1024 pairs, but it was something like that.

That limit in the number of elements is what I was concerned about.
Because I wrote the script without understanding enough about Ai's brush.
In above script, I'm adding 2 elements to the tabletData.
If there's that limit, it may cause an error or an unexpected result.
So I tried drawing very long strokes with a brush.
But they had only 50 or so tabletData-elements at a maximum.
It seems that there is no need to worry ...?

By the way, to tell the truth, I thought to create that kind of brush tool as a plugin recently.
I read Ai-SDK's document to find it's too difficult to me for now, but I understood the brief of the tabletData.
And also I found that Ai-SDK's document is even very helpful to understand Sg's functions.

Hiroyuki- excellent stuff! I've been trying to work out a similar tool for a while, and looks like it will cut the work needed in half! I can't seem to get it to excecute properly though. Each time I run either script above, it returns null on the getTabletData, or seems to crash on the setTabletData function. Does the path need to be created with a tablet to have the tabletData array populated?

It seems that only a path drawn with Brush Tool and a calligraphy-brush has a tabletData. It need not necessarily a stylus pen.

The script I uploaded first (on 04.02.07) was just a simple sample and had no error handling. It causes a crush if a path has no tabletData.
It was too careless, indeed.
I added the lines marked '#' to avoid errors to the first script.

Also, I added an additional process at the end of it.
It seems that it's possible to apply a tabletData to a path that has no tabletData.

The second script does nothing if a path has no data.
But both of these do not work nicely if a path is not drawn with a stylus pen.
I will fix the second one about this issue.

It seems that the data added in the second script wasn't applied to the stroke.
I made a quick-fix on 'taper.js' to avoid an unexpected result with strokes that wasn't drawn with stylus pen.
But it needs other kind of tweak to work nicely in any case.

I've been experimenting with this for a few days, and have gotten mixed results with brush paths created without a tablet. I'd like to do some more research, but lack the data... Can someone post or email a CS2 .ai file containing a tablet-drawn brush path? One saved as a plain file (without pdf compatability and compression) would be preferred. Any help would be appreciated!

I don't mean to beg, but I really would like to see how the tabletData array is structured. If a full file is out of the question, then just the path's data along with it's tabletData would be just fine!