I have discussed at various other times the importance of the corporate element of our worship. This is also true in communion, as Phillips notes in these quotes:
“Christians of every kind need to beware of pietistic individualism, and this no less true at the focal point of worship than at any other place in the Christian life.” (26-27)
“Holy Communion is surely always falling short of its true purpose if it fails to produce some sense of solidarity with our fellow worshippers.” (28)
Part of the point according to 1 Cor 11 is awareness of our fellow members of the congregation. We often miss this in our individualistic practice of shutting everyone else while we individually commune with God. This is why at our church we have gone to keeping our eyes open and drawing attention to the fact that we partake of these elements together, as one body reaffirming faith in this one Savior who has made us all part of His one Body.
J.B. Phillips, Appointment With God: Some Thoughts on Holy Communion