Yes....Paul arrived in Jerusalem [Acts 21:17] after returning from his third missionary journey. His arrest had been foretold [Acts 21:10-12] and his presence in Jerusalem caused a stir [27-40]. He appears before the crowd and with permission from the commander explains his case. The commander's entire demeanor changes when he learns of Paul's Roman citizenship [Acts 22:26-29].
Paul then appears before the Sanhedrin and a plot develops to kill him but the commander, knowing Paul to be a Roman citizen, called two of his Centurions [Acts 23:23] and a detachment of 200 soldiers....seventy horsemen...and 200 spear men to escort Paul to Caesarea! Not only does he get an escort fit for a king....but gets horses(plural) to ride along the way. Being a Roman counted for something!
On to the Governor (Felix) and Paul is then kept under guard in Herod's palace [Acts 23:35]. The trial before Felix begins and Paul again states his case. This goes on and on and on....two years pass [Acts 24:27]! it is now about 59 A.D. and Paul meets the new Governor, Porcius Festus. Festus learns of the details of the case, has Paul brought before him [Acts 25:6] gives Paul the option of a trial in Jerusalem and Paul makes his appeal to Caesar's court and Festus obliges (verse 12).
Paul appears before King Agrippa after the King had discussed Paul's case with Festus. Agrippa makes his profound statement in [Acts 26:32] "This man could have been set free....if he had not appealed to Caesar." Little did King Agrippa know that both God and Paul wanted Paul to go to Rome [Acts 19:21]. It is late 59 A.D./ early 60 A.D.
After many perils on the voyage to Rome (shipwreck etc.) he finally arrives in Rome [Acts 28:14-16] where he meets some Christians who have come to greet him....some Jewish elders who have not heard much about this new sect [verses 15-22], want to hear more....some believe and others don't. He preaches the gospel from his own home unhindered [Acts 28:30-31], writes four maybe six epistles and finally is released after two years. It is now 62 A.D. and he has not spoken of..... or seen Peter since the Jerusalem council 49 A.D.
During the first imprisonment (60/62 A.D.) evidently Timothy was with him [Hebrews 13:23] as he salutes everyone jointly with Timothy in Philippians 1:1 but during the second he writes to Timothy, telling him to go get Mark and bring to him his scrolls and parchments. He evidently had left them at Troas during his arrest and subsequent second Roman imprisonment, having no time to collect his personal belongings. In [2 Timothy 1:4] he recalls Timothy's tears....probably as he is being dragged off by Roman soldiers. The second imprisonment is thought to be about four years later (66 A.D.)
While Paul had to remain in Rome, Peter would not have had to and no doubt would have left there at that time, seeing the trouble that was coming.
He would have left Italy, gone back to Asia Minor, then on to Parthian Babylon where he would then write his first epistle in 65 AD, and probably his second in 66AD, probably dying there shortly thereafter.
The apostles were missionaries, not settlers or bishops, and Peter and the other apostles lived by the words of Jesus recorded in Matthew 10:23: "But when they persecute you in this city, flee into another", until, of course, their end had come.