Not quite. Peter focused his attention on the church in Judea, centered in Jerusalem while Paul turned his attention to taking the Gospel to the Gentiles. This was an agreement between the two. Peter gave unrestrained authority to Paul in that arena as many of the Jews in the Holy Land did not consider Paul to have any Apostolic authority. The prominence Paul has is due to his letter writing and establishing numerous churches along his missionary trail.
Paul already had authority to preach as one sovereignly chosen by Christ who Himself revealed to him the gospel.
But I certify you, brethren, that the gospel which was preached of me is not after man. For I neither received it of man, neither was I taught it, but by the revelation of Jesus Christ. (Galatians 1:11-12)
But when it pleased God, who separated me from my mother's womb, and called me by his grace, To reveal his Son in me, that I might preach him among the heathen; immediately I conferred not with flesh and blood: Neither went I up to Jerusalem to them which were apostles before me; but I went into Arabia, and returned again unto Damascus. (Galatians 1:15-17)
Paul prefaces his testimony to Galatians - who are being seduced by Judazers at conflict with Paul - with such words of his sovereign commission by the Lord and revelation from Him, and adds that "after three years I went up to Jerusalem to see Peter, and abode with him fifteen days. But other of the apostles saw I none, save James the Lord's brother." (Galatians 1:18-19) Thus Paul was preaching with authority for years before going up again to Jerusalem (not Rome).
From this it is surmised that since many rejected Paul as an authority, which could overthrow the faith of the churches he founded, then manifestly obtaining the "right hand of fellowship" (Galatians 2:9) from those who "seemed to be pillars" was warranted. And in which Peter is not presented as the supreme head, but is named second among 3, after James and before John.
But holy Peter (a far far better man than I) has distinction as being the only apostle charged with misleading the brethren and being publicly reproved for so doing by the apostle you have him singilarly giving authority to, to do what he already was.