In addition to browsing and playing music on the machine where it runs, iTunes has two major networking features: 1) To act as DAAP (the iTunes music sharing protocol) client to browse/play music on other machines, and 2) To act as a DAAP server to make music on its host available for clients elsewhere. Most of the visible UI of iTunes is the local/client part. The server part is just the "Share my library" feature and has no UI beyond what's needed to turn it on.
Banshee replicates much of the iTunes local/client functionality. You can browse and play locally or access shared music from any DAAP servers on your network (theoretically; it seems to have problems talking to iTunes 7 and later). Some versions of Banshee had DAAP server (sharing) capability but that seems to have been removed in more recent versions. This is probably because there are already very good DAAP servers for Linux (many people in this forum use mt-daapd) and the Banshee folks prefer to focus on the client/UI parts.
From this perspective SoundBridge is just another DAAP client. It can access DAAP servers such as a shared iTunes (Mac or Windows) or mt-daapd/Firefly or similar on Linux or whatever. But two clients don't have much to do for one another, so having Banshee running on your Linux box provides nothing useful for your SB.
So if your music is stored on your Linux machine, run mt-daapd on it and your SB and iTunes machines should be able to use it. Banshee on the same machine can either read the files directly or access them as shared music from the mt-daapd server. If your music files are on the Mac, turning on sharing makes them visible to the SB and could theoretically, but probably won't, make them accessible from Banshee. An alternative is to run a Firefly (mt-daapd) server on your Mac, which should work with both the SB and Banshee/Linux (and you can turn off iTunes sharing).