Unfortunately there's no way to actually vary the stroke width. This was a limitation of SVG1 - I think that variable width strokes made it through for SVG2, but may have been punted to 2.1, and certainly isn't supported by Inkscape yet.
The only workaround is therefore not to draw the outline as a stroke, but to draw it as a complex path in its own right, where the black of the outline is actually the fill on an unstroked path. This is what LPEs such as Taper Stroke and Power Stroke do. As Druban said, you need the fill to be a separate object - although it is possible to use the "Clone Original Path" LPE to achieve the same effect. See part 47 of my tutorials, here:
viewtopic.php?f=6&t=11981 but also look at parts 66 & 67 as this functionality was changed (some might say "broken") in recent releases.
One other workaround is to apply a gradient to the path such that the ends appear to fade out. This can be a bit tricky to get right, depending on how the ends of your paths are positioned, as you can only use a single gradient that has to accommodate both ends without affecting the middle. It doesn't actually thin the stroke, but can give a similar visual effect, particularly for thin strokes.