1 Cor. 10:16, "The cup of blessing which we bless, is it not the communion of the blood of Christ? The bread which we break, is it not the communion of the body of Christ?" Communion is the Greek word, "koinonia" (Strong's word 2842). Strong's defines it as "partnership, that is, (literally) participation". The Apostle Paul is explaining to believers when they eat the symbols of our Lord's sacrifice, the unleavened bread and wine, they are remembering Jesus' sacrifice and applying it to themselves. However, since believers are also termed the body of Christ, when we partake we are also celebrating our union with the rest of the body of Christ. Colossians 1:24, "Who now rejoice in my sufferings for you, and fill up that which is behind of the afflictions of Christ in my flesh for his body's sake, which is the church."
This beautiful picture is best experienced with other believers. Jesus said "For where two or three are gathered together in my name, there am I in the midst of them." (Matt. 18:20) Let the consecrated followers of Christ gather together (physically) to celebrate this unity. However, due to some restrictive situations (like illness, war, isolation, etc.) it may not be possible to partake of the Lord's memorial with other believers. In such a case, the scriptures do not forbid a personal celebration at home. This option is available, but certainly not preferable.