Well, if you're conducting research using the papyri you're probably not going to get to handle the papyri themselves. Rather, you would probably be using high resolution images given the fragility of the source material and the fact that they're held in different collections. Additionally, one would not work with the Oxyrhynchus Papyri exclusively. Rather, one would probably draw from evidence there, as well as other early Christian writings. These people are likely classicists or scholars in religious studies departments. All would be familiar with Greek (and Latin, though it's not strictly necessary for this), as well as Coptic, Syriac, Hebrew, and others.

That being said, there is a crowdsourced project to get lay people involved with transcription: https://www.ancientlives.org/. The website looks a little sleepy, and i think it's under construction, but you might look into that.