How to produce a reproducible column of random integers in SQL

I have a table of patient, with a unique patientID column. This patientID cannot be shared with study teams, so I need a randomised set of unique patient identifiers to be able to share. The struggle is that there will be several study teams, so every time a randomised identifier is produced, it needs to be different to the identifier produced for other studies. To make it even more complicated, we need to be able to reproduce the same set of random identifiers for a study at any point (if the study needs to re-run the data for example).

I have looked into the RAND() and NEWID() functions but not managed to figure out a solution. I think this may be possible using RAND() with a seed, and a while loop, but I haven't used these before.

Can anyone provide a solution that allows me to share several randomised sets of unique identifiers, that never have the same identifier for the same patient, and which can be re-run to produce the same list?

Thanks in advance to anyone that helps with this!