What is the nature of the "Communion of the Saints"? In the Episcopalian Prayers of the People, Rite I, we pray as follows:


  In the communion of [__ and of all the] saints, let 
  us commend ourselves, and one another, and all our life, to 
  Christ our God.


As such, we have at least one Protestant (sort of) group that acknowledges the concept of the communion of the saints.  That said, there isn't a lot of understanding of that loaded term.

This term, of course, harkens back to the Apostle's Creed which states:


  I believe in the holy catholic church: the communion of saints: the forgiveness of sins, and the life everlasting.


The question, however, is what is the nature of this communion?  Is it a fixed gathering of all believers at one time in one place, or is it like an earthly institution, like say, the Boy Scouts, where the membership changes over time, but there is always a membership.
From a reformed background the communion of the Saints is a mystical union every believer has with one another due to our common union in Christ.  This does not necessarily imply we are actually 'experiencing others' that we have never met.  It means we are all united in Christ as a new creation just as all humanity was once united in Adam.  By being members of the same mystical body, we share the same nature and life, which is communion with God.


  that which we have seen and heard we proclaim also to you, so that you too may have fellowship with us; and indeed our fellowship is with the Father and with his Son Jesus Christ.  (1 John 1:3, ESV)


Simply put, the communion of the Saints is the church.  Not visible, but invisible.  The invisible church refers to those persons who are redeemed, regenerate and spiritually united with Christ by sincere faith.  It is not dependant upon a denomination, building or sacrament. Faith alone unites a person into Christ as members of his body.


  For through him we both have access in one Spirit to the Father. So then you are no longer strangers and aliens, but you are fellow citizens with the saints and members of the household of God, built on the foundation of the apostles and prophets, Christ Jesus himself being the cornerstone, in whom the whole structure, being joined together, grows into a holy temple in the Lord. (Ephesians 2:18-21, ESV)


Of course the church, or Christ's body includes those in heaven and on earth. We are all mystically joined into one body with the same desire, love, hopes etc. by the same Spirit.