We ran into the same issue, it's more pronounced if your community is full site SSL enabled (HTTPS). Most browsers won't allow the CSS to load from your local machine as they see this as insecure.
The workaround for this unfortunately is a bit involved and requires you setting up an Apache Web Server on your local machine, creating symbolic links to the required directories in the SDK files and configuring SSL for the local web server.
For example instead of http://localhost:9000/skin_name/skin.css our staging environment now points to https://localhost/skin_name/skin.css wich works.
Can you confirm if your staging site is https:// enabled?
