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Amazon has something somewhat related long ago called "Someone like you" that would show you books purchased by people who had similar previous reading purchases. It was odd to see someone had purchased the same combination of OpenGL books, cookbooks, and sociopolitical books as me. The recommendations that resulted were pretty good but I suspect privacy concerns ended it.

I'm not a Goodreads user, but this is something you can definitely do with Librarything. For any user in the system with a public collection, you can find the intersection of your books and sort by the number of members who own them (as well many other things). The interface is a little wonky, but very powerful.

> for me and a friend, show me the rarest book we have read in common

Got to your Friends page (https://www.goodreads.com/friend), click on "Compare books" in front on the friend with which you want to compare. You will see the list of books you have in common sorted by inverse popularity.