In natbib there is \citetext{priv.\ comm.} which is used to add a textual citation to the bibliography that doesn't have a key associated with it.
I don't see a way to get something like that in org-cite, since it seems that a key is always required.
This isn't currently recognized as a cite, but something like this seems like a reasonable solution to me.
[cite/text:@ private communication]
John
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Professor John Kitchin (he/him/his)
Doherty Hall A207F
Department of Chemical Engineering
Carnegie Mellon University
Pittsburgh, PA 15213
412-268-7803