The short answer is that output sharpening should be done after the image has been resized. If you take a full size file that has been sharpened for printing and downsize it to web size, the result will generally be a bit soft. Depending on the content and the intended purpose, you may consider the result acceptable, but it will not be optimal.
Sharpening a digital file really consists of creating very narrow bands of dark and light to enhance the contrast along the edges of objects in your image. The trick is to have those narrow bands created in a size and intensity so that when the final image is displayed, the eye of the viewer will see only the appearance of a contrasty edge to the object, not the appearance of actual dark and light bands. So if the bands are too wide or intense, the viewer will see a halo effect. If the bands are insufficient, the image will appear soft.
Imagine a full size file (say 12MP) with a lot of detail, like a silhouette of a tree branch. If you reduce it to web size (say 500KP), you will lose a lot of detail--many fine twigs will not be visible. If the image had been sharpened at full size, then some of the detail you will lose will be the fine variation in light and dark that was created for proper sharpening at print size.
The different resizing algorithms (bicubic sharper and bicubic smoother) will tend to either preserve detail or preserve more natural looking edges, but I don't think they are a substitute for output specific sharpening.
For a much better explanation of all this, go to the Pixel Genius web site and download the user guide to the PK Sharpener software (a product I highly recommend). The examples are very clear.