No. Paul was not writing to the same audience as Peter, and vice versa.
Peter had the ministry to the circumcision, and wrote to the Jews of the Diaspora in Asia Minor.
Paul had the ministry to the uncircumcision--that is, the Gentiles--and (except for the letter to the Hebrews, which may be attributed to him) there were different audiences than Peter's, mostly Gentiles, whose cultural ignorance of Jewish/Christian theology needed to be addressed, and many different matters were discussed, ones which Jews already were quite familiar with and for them needed no discussion.
I believe your comment on this is quite naive, and needs correcting.
What could be less sentient and more naive than trying to say that Peter was ministering to the circumcision when he wrote to the gentile northern Israelites in Asia minor?
He opens his epistle declaring that he was writing to the “strangers.” Checking Strong’s you will find that those strangers were the lost sheep of the dispersion, the very same people to whom Paul wrote most of his letters.
.