Consider this passage from 1 Peter:
As newborn babes, desire the sincere milk of the word, that you may grow thereby, now that you have tasted that the Lord is gracious. As you come to him, the Living Stone, rejected by men but chosen of God and precious, you also, as living stones, are built up a spiritual house ... (1 Peter 2:2-5)We find the same thoughts together in 1 Corinthians chapter 3, only a few verses separated from each other, in a passage where Peter is mentioned:
I have fed you with milk, and not with meat: for hitherto ye were not able to bear it, neither yet now are you able. (I Cor 3:2)In this same passage, Paul discusses different leaders in the church building on each others' work. By now, that's enough to make me wonder whether Paul had read Peter's first letter ... or whether that was just a common preaching point of Peter's.
For we are laborers together with God: you are God's workmanship, you are God's building. According to the grace of God which is given unto me, as a wise master builder, I have laid the foundation, and another builds on it. But let every man take heed how he builds. For no man can lay a foundation other than that which has been laid, which is Jesus Christ. (I Cor 3:9-10)
Do you not know that you are the temple of God, and that the Spirit of God dwells in you? (I Cor 3:16)
... Whether Paul, or Apollos, or Cephas, or the world, or life, or death, or things present, or things to come; all are yours. And you are Christ's, and Christ is God's. (I Cor 3:22-23)
I also find it interesting that in 1 Peter, he mentions the name of his scribe: Silvanus (often translated Silas). Silvanus is also mentioned in Paul's letters. He is mentioned in 2 Corinthians to the same people that Paul was addressing earlier with his comments about Peter and about building on each others' teachings. He is also mentioned in both letters to the Thessalonians. Both the church in Corinth and Silvanus in particular were familiar with Peter. So we also have some ways in which Paul might have come across Peter's letter, if Silvanus had kept a copy, or one of the churches had obtained a copy. That doesn't make it certain that Paul had come across a copy; it's still possible that it was passed along through word-of-mouth. But it is enough to raise the possibility that Paul had seen Peter's letter.