Q:

How do "Friend of P" and "Friend of P's" mean the inverse of each other? 

I abbreviated Peter as P in the title for brevity; feel free to abbreviate thus.  Please explain user Ed Staub's answer below, since it only seems to present this stance?  To me, both 1 and 2 mean that Joe and Peter are friends with each other.  Yet what possible reasons exist to explain the differentiation below?  

To me, "Friend of Peter" and "Friend of Peter's" mean the inverse of each other. 
In "Joe is a friend of Peter", Joe is the active person in the
  friendship - it describes Joe's active relationship to Peter.  Peter is
  one of the people Joe expresses friendship toward. 
In "Joe is a friend of Peter's", Peter is the active person in the
  friendship - it describes Joe as being the object of Peter's
  friendship.  Joe is one of the people Peter expresses friendship
  toward. 

A:

When we say that Joe is a friend of Peter we are making a statement that Joe and Peter are friends.   
When we say that Joe is a friend of Peter's we are identifying Joe, who is unknown to the listener, as a friend of someone whom the listener does know, namely Peter.   It has nothing to do with a putative "active person in the friendship". 
A friend of mine, Mary, arrives for an impromptu visit.   She is introduced to a stranger, Henry.  After Henry leaves the room, Mary asks, "Now, who is Henry?"   I reply, "Henry is a friend of Jane's".    Mary knows who Jane is.