The issue with P2P is not the product you use.

The issue is that the peer you are connecting to decides which port they want you to connect to them on. So if they happen to be hosted on whiterabbit.net and they tell you they are listening on port 110 or 25 or 143 then that would be the connection you make and avast will scan it. The Internet Mail scanner icon will appear in the tray and the mouseover will tell you it is connected to whiterabbit.net.