Can a USB port passively 'listen only'?
USB CDC serial is quite a complicated protocol. Even if you are only sending serial commands one way, there is USB traffic going backwards and forwards all the time. Even when the virtual serial bus is quiet, the usb host is continuously asking the usb device "do you have anything to send" and the device says "no".

Consider sending the phrase "hi" to an arduino, and getting "hello" back. On the USB bus, it will look something like this:

Host: Do you have a message?
Device: No.
Host: I have a message for you
Device: OK
Host: hi\n
Device: OK
Host: Do you have a message?
Device: No.
Host: Do you have a message?
Device: hel
Host: Do you have a message?
Device: llo\n
Host: Do you have a message?
Device: No.
etc. etc.

These messages will also be mingled in with messages sent to other devices on the same hub (even if you don't have a USB hub in the system, there may well be one inside the computer. So you could easily see messages to your mouse and keyboard mixed in). 

The USB protocol is extremely complicated, so splitting it in the way you mention is not going to be practical.  You can "sniff" it though. If you connect the D+ and D- lines, it is possible to see the traffic on the bus.  As long as the sniffer doesn't try to manipulate the D+/D- lines, the USB bus will keep working. The sniffer would then need to identify which messages were relevant and decode them. Building a sniffer out of an arduino would be difficult but probably not impossible. It almost certainly isn't the best solution for your problem.