In this context people sometimes proceeded quite liberally in building the chain of tradition...Then, as now, historical thinking was overlaid with wishes.
The idea that both of them [Peter & Paul] first came to Rome after the church had already existed there for a longer time had no place in early Christian thinking, which in this case wanted to forge a connection between something they knew and the earliest and best-known men whose names they knew.
In the first century and the beginning of the second, the Roman church was led by a college of presbyters, as we learn reliably from 1 Clement which we have frequently mentioned. We can no more speak about an apostolic succession, by which Peter passed on the episcopal office by a laying on of hands, than we can about many other things. This idea was a product of the second century when the idea of apostolic succession inevitably developed from the concept or requirement of apostolic tradition. Both existed only after the second half of the second century."
K. Aland, A History of Christianity (Fortress 1985), 1:118-120