The basis seems to be that you can only transfer books between kindles registered to one Amazon account. So: you deregister your kindle, then re-register it to my account, calling it, say, Ariadne's Kindle2, download the books we want to share, then register your kindle back as yours again.
What we have since found is that now, when she goes to my library, she can click the Actions button and select the delivery option, to Ariadn's Kindle 1 (mine) or 2 (hers)
So - we just register our kindles to each other's accounts and download the books we want to share, and then register back to the original.
It does, of course, mean that you have to give the other person access to your Amazon account! And I would advise doing it sitting next to one another on the settee.