If I give my IPOD Nano to another person, can they reset it so they can use it on their computer or do I have to do that on my computer?

You have to un-authorize the iPod first, then he can use iTunes to restore the iPod and re-authorize the iPod, And then he can use it on his computer.

Either one of you can do it, just Restore it in iTunes and don't sync to it, if you are doing so on your computer, or sync it if it is on the other persons. When they restore it then it will remove all of your data and install theirs. You do not authorize or unauthorize iDevices only computers.