I think that if we use the word "eucharist" instead of "communion", we get a different perspective. To me, the sacrament is much more than fellowship with members of the local church, or our denomination. If the sacrament were about fellowship, then one could decide whether one wanted it to be fellowship with the local church, with those who they accepted as being part of the universal church. Even under this paradigm, I think that Anglicans have it right. All validly baptized are part of the body of Christ.
The eucharist is not fellowship. The word means thanksgiving. It is about the invisible body of Christ uniting with the visible body of Christ, with the Cross and with all eucharists since the beginning. We re-live (the Jewish concept of remembrance) the cross. Jesus is NOT crucified again. We present ourselves as an acceptable and living sacrifice and we are all made one with His crucifixion.
For me, we receive the body, blood, soul and divinity of our Lord Jesus Christ in the Eucharist. For me that is what the prayer of consecration is about. Indeed, we are what we pray, and this has been the prayer from the beginning. HOWEVER, the mechanism, the transformation is NOT all that important to me. We all become on with (commune) with the body and blood of Jesus. Lutherans have their paradigm, Catholics have theirs, Othodox have their paradigm, protestants have theirs. Anglicans and Methodists generally ignore the theology.
What is critical is that after we receive, we are united with Christ and the Cross. We have received our spiritual food at the Lord's table, at the thanksgiving feast.
To me it is abhorrent to refuse a seat at the Lord's Table to baptized Christians who are visiting our local churches. It is not our Table. It is the Lord's Table, the thanksgiving feast.