Christians universally agree that the Lord’s Supper was instituted by Christ and should be observed as an ordinance in the church by His followers. It was to the Corinthian church that Paul wrote instructions concerning the Lord’s Supper (1 Corinthians 11:23-26). Paul later wrote to Timothy about the qualifications for church leaders, bishops/elders and deacons (1 Timothy 3:1-13). In the original language, the word “deacon” comes from a verb that means "to serve," probably in the sense of waiting on tables, but it also came to be used to signify a broad range of service in the church. Because of the connotation of table service in the word “deacon” and the centrality of the Lord’s Supper in the worship of the early church, there is strong indication that serving the communion elements was an important function of deacons.
From this we can conclude that designated church leadership conducted the Lord’s Supper in the early church; however, there is no Scripture specifically given with "how to" instructions. Therefore, it would seem reasonable for the leadership, if there were an insufficient number of deacons present, to appoint laymen to serve.
More important than who serves communion is the attitude with which it is both served and received. First Corinthians 11:27 says that those who take the elements in an “unworthy manner” are guilty of sin against the body and blood of Christ. An “unworthy manner” can mean the taking of the elements by those who do not belong to Christ or taking them in a flippant or irreverent manner. It can also mean using the ceremony as a means to be seen before men to be exalted by them. Verse 28 gives the criteria for both serving and participating in the Lord’s Supper. We are to examine ourselves before we partake and be sure our hearts are right before the Lord. Then both the servers and the receivers can be sure of pleasing God when they participate in His communion.