Sync playlist to iPod on other networked iMac
iPod/iPhone syncing only work with local files, meaning you have to copy your files to your kids computer or have them sync at yours (yes, it's a pain both ways).

Home Sharing somewhat helps here because it allows to copy purchased stuff automatically as explained in HT3819:


  You can also choose to have iTunes automatically import purchases you make on any of your Home Sharing computers to your own library. To do so, select the library and use the Settings button at the lower right of the iTunes window to check what you'd like copied.