Because the reading list interface is currently in a transition phase, there is some functionality that is missing from the new interface.
If you need to assign the list to a new owner, you need to go to the 1.0 version of the list. From the list view, click on the Beta button - this is highlighted in the screenshot below.
1) Go to your reading list and click the Beta button (as shown above) to open the version 1.0 interface.
2) Click on Edit to open the drop-down list and select Assign List Owner.
3) A pop-up window will appear which will show the current list owner. You can then search for the person you want to assign the list to or click on the assign to me link to assign to yourself.
4) Click Assign with publisher role to give publisher rights to the new list owner.