In Errata, Redjack asked for clarification of an apparent conflict. SR4A 218 says
The routing functions of a device are handled by a separate component of hardware than the other functions of the device. This makes the routing process invisible to the user, and allows the device’s node to connect to the Matrix even when it is operating in Hidden mode
while Unwired 54 says
Due to the mesh-network nature of the Matrix, every wireless node can function as a router and will do so if not in passive or hidden mode
Redjack thinks the Unwired definition should apply. After thought, I disagree. The reason is that passive and hidden nodes can still be seen with matrix perception.
If'n I were to explain what's going on, I'd do it this way. (My opinion, not RAW or even FAW (Fluff as written).)
The settings don't touch the router. The router always receives, and always forwards. Forwards can go to any other router, and any addressed to its node are forwarded to, well, the node.
What the settings affect is the node. In active mode the NODE sends its access ID and a supplemental ID package; public profile stuff. Other nodes receive the public profile stuff and send material related to the public profile. (Spam, of course, is "related" to any/all public profiles. sigh.) One critical element of the public profile is the persona ID.
In passive mode the node sends /just/ the access ID. The spammers still send to the access ID, but since they don't also have the personal ID they're dropped. Essentially, they're packets addressed to occupant.
In hidden mode the node doesn't even send the access ID. It receives, it forwards. If the access ID is in the packet it will forward it to the node. This, by the way, is why a team which has set all its comms to hidden is still in communication. They know their teammates' access IDs.
Are there weaknesses? Yep. Does knowing the answer make or break the game? Nope. But it's still my working hypothesis for how it works.