Much to my dismay, Silverlight 4 going out of browser with GZip compression just isn't supported.  With Silverlight in the browser, the browser will likely add the Accept Encoding header with gzip (and probably deflate) for you.  Then the browser will also decode the optionally gzipped data automagically.

I guess that's why there is no GZipStream in Silverlight at all :)

Fortunately, I can use my own project SharpCompress to help me out.

Using the WCF WebAPI, I compress everything on the server and inspect the content on the Silverlight client then just ungzip.