Apple - Can apps purchased from one Apple ID be shared between multiple iTunes libraries?

Unfortunately the only way to share apps is to use the same Apple ID for purchases on every device.

You can use different IDs for iCloud, but you need to sign into the App Store with the same ID.

If you're signed in to the App Store with the correct ID you can download previously purchased apps for free, each person haven their own computer and iTunes library doesn't matter.

With Home Sharing and Authorizing computers every person can have their own account and still use apps purchased by other family members.

Home sharing will allow you to stream, browse and copy content, including apps, between machines. You turn on home sharing one each machine under iTunes's Advanced menu. You use the same Apple ID on each machine. It doesn't matter which Apple ID just as long as it's the same one, all this ID is for is to make all of the machines known about the Home share you're setting up. Apple has instructions online: http://support.apple.com/kb/HT3819.

Authorizing each computer lets you use protected contact on that machine, including syncing apps to associated iOS devices. You'll need to authorize each computer with each of the accounts you will be purchasing apps with. For example, my wife and I both need to authorize our son's machine. Again, apple has instructions online: http://support.apple.com/kb/ht1420 .

There is a limit to how many machines you can authorize.

Once you do both these things each person can use Home Sharing to look at the apps on another machine (its listed in the iTune's sidebar), import those apps to their own machine and those apps will sync to their iPod.

The one down side is app updates. You will need to reenter your password on each iPod in order to download an update. Alternatively, the original purchaser can download the update and the others can grab it again using Home Sharing.