You can sync an iPod with only 1 "iTunes library" at a time. It will not sync with any other library, unless you register it with the other library, which will clear the songs and reset the iPod.
You can however, view the songs on the iPod when it is connected with another PC/Mac, but you can't copy songs from the iPod onto the computer. When it comes to the iPod shuffle however, it will not even show the shuffle on the other PC. It will only offer the option of "erase and sync", otherwise you can't use it on another PC.
You can however, use all the iPods to copy any DATA files while using it as a USB Mass storage drive, like a pen drive. You can copy to and fro and between computers, no problem.
Macboy could be wrong.