Paul's letters were delivered to a wide range of locations. His writings were sent to Thessalonica and Philippi in Macedonia, Corinth in Achaia, Ephesus and Colosse in Asia and Rome in Italy. Paul's letters were also addressed to the churches in the province of Galatia (Pisidian Antioch, Iconium, Lystra and Derbe), the island of Crete and Jerusalem in Judea.
The below list, in chronological order of Apostle Paul's letters, reveals (where possible) those who helped deliver his words to various Christians in the Roman Empire.
Thessalonians
It is unknown who delivered both of Apostle Paul's letters to those in Thessalonica.
It is possible that 2Thessalonians was sent to the church either by Silas or Timothy. Both men, when the letter was authored in 51 A.D. from Corinth, were with Paul in the city (1Thessalonians 1:1, 2Thessalonians 1:1). They also had experience serving Christians in Thessalonica (Acts 17:10, 1Thessalonians 3:1 - 5).
1Thessalonians, however, seems to rule out either Silas or Timothy taking it to the church due to what is said at the end of the letter.
I (Paul) adjure you by the Lord that this epistle be read to all the holy brethren (1Thessalonians 5:27).
The command of 1Thessalonians 5:27 would be unnecessary if Silas or Timothy went with the letter. They were Paul's trusted helpers who would have insured it was read to the church.
Galatians
The Bible is also silent regarding who delivered the book of Galatians. Timothy, Erastus, Gaius, Aristarchus or Titus (Acts 19:22, 20:4, Galatians 2:1), who were with Paul when he wrote the letter from Syrian Antioch, may have been used to take it to Galatia.
Corinthians
It was Titus (and an unknown person, 2Corinthians 12:18) who left Ephesus to deliver 1Corinthians (Life and Epistles of St. Paul, chapter 15, page 418). He was also likely the person who took 2Corinthians to Corinth (International Standard Bible Encyclopedia on Titus).
Romans
The book of Romans definitively tells us who the Apostle Paul used to deliver his letter to Rome.
To the only wise God, through Jesus Christ, be the glory into the ages of eternity. Amen. To the Romans, written from Corinth; delivered by Phoebe, a deaconess of the church in Cenchrea (Romans 16:27).
Phoebe, who was visiting Corinth, was a female leader of the nearby church in Cenchrea. Paul took advantage of her upcoming business trip to the empire's capital (Romans 16:1 - 2) to have her take his letter to the city's Christians.
Hebrews
It was Timothy, Paul's close friend and faithful fellow co-worker, who delivered his letter to the Hebrews.
May God’s grace be with all of you. Amen. To the Hebrews, written from Italy, delivered by Timothy (Hebrews 13:25).
The book was addressed to Jewish converts to Christianity primarily living in Jerusalem and Judea (Jamieson-Fausset-Brown Commentary and Johnson's Notes on the New Testament on Hebrews 13:25).
Ephesians
Paul's letter to the church in Ephesus is considered one of his "prison epistles" as it was written during his incarceration in Rome. Other books in this set include Philippians, Colossians and Philemon.
Ephesians reveals that Tychicus was tasked with delivering it.
Now that you may also know the things concerning me and what I am doing, Tychicus, a beloved brother and faithful servant in the Lord, will make everything known to you . . .
Grace be with all those whose love for our Lord Jesus Christ is uncorrupted. Amen. To the Ephesians, written from Rome, delivered by Tychicus (Ephesians 6:21, 24).
Philippians
Philippians, like Ephesians, was considered one of the Apostle Paul's prison epistles.
Epaphroditus was lovingly referenced by Paul as his brother, fellow worker and soldier (Philippians 2:25). After his arrival in Rome he stayed in the city aiding the apostle in his missionary work.
During Epaphroditus's zealous efforts to help the apostle he got sick and nearly died (Philippians 2:27). His merciful healing by God motivated Paul to send him back home to Philippi with his letter to the church.
But I felt it necessary to send to you Epaphroditus, my brother and fellow worker and soldier, but your messenger and minister for my need, since he was longing after all of you, and was deeply depressed because you had heard that he was sick . . .
May the grace of our Lord Jesus Christ be with you all. Amen. To the Philippians, written from Rome, delivered by Epaphroditus (Philippians 2:25 - 26, 4:23).
Colossians
It is clear from the letter to Colosse who were the two people who delivered it to the church.
The salutation of Paul by my own hand. Remember my bonds. Grace be with you. Amen. To the Colossians, written from Rome, delivered by Tychicus and Onesimus (Colossians 4:18).
Philemon
The subscription for the book of Philemon tells us who delivered this personal letter from Paul.
The grace of our Lord Jesus Christ be with your spirit. Amen. To Philemon, written from Rome, delivered by Onesimus, a servant (Philemon 1:25).
1Timothy
The Apostle Paul wrote his first letter to Timothy in 63 A.D. after he was acquitted and released by the Romans in Rome.
1Timothy is one of the few Paul letters where it is unclear who delivered it, from Nicopolis, to Timothy in Ephesus. Luke, who was with the apostle during his imprisonment, may have still been with him and could have carried the letter to its destination.
Titus
Paul, after he was released from Roman imprisonment, wrote to his friend Titus in 63 A.D. His letter was written from Nicopolis in Macedonia where he planned to spend the winter (Titus 3:15).
It is highly likely that either Artemas or Tychicus, whom Paul was sending to Titus, delivered his letter to the island of Crete.
When I send Artemas to you, or Tychicus, make haste to come to me at Nicopolis; for I have decided to spend the winter there (Titus 3:12).
2Timothy
In January or February of 68 A.D., just before his martyrdom, the Apostle Paul writes his last Biblical letter. Its closing subscription doesn't record who delivered it to his close friend.
The Lord Jesus Christ be with your spirit. Grace be with you. Amen. The second epistle to Timothy, who was chosen first overseer of the church of the Ephesians, written from Rome when Paul was brought a second time before Caesar Nero (2Timothy 4:22).
2Timothy, however, does mention that Paul's friend Luke was with him. It also reveals that several brethren, likely from the local fellowship, also visited the apostle.
Only Luke is with me. Get Mark and bring him with you, because he is profitable to me for the ministry of the Word . . . Eubulus salutes you, and Pudens, and Linus, and Claudia; and all the brethren salute you (2Timothy 4:11, 21).
It is one of the people, mentioned above, who likely insured Paul's letter was delivered to its intended destination.
Conclusion
The Bible reveals the names of those who either delivered Apostle Paul's letters or were available to do so. These included individuals such as Timothy, Titus, Luke, Phoebe, Tychicus, Epaphroditus, Artemas and others. Their efforts insured God's people received the instruction they needed and that Paul's words would continue to be read for generations.