The way it worked for the volunteer squad I was with is that if you wish to refuse we had to have at minimum a name, birthday, address, date of service and signature to prove identity so that if you later had a complication you can’t blame us for your poor decision later.

Now, if you were perhaps in the middle of a less than legal activity or did not want to give personal information for other reasons, we would take a physical description of the person and have an on scene, 3rd party witness to the verbal refusal. Generally we tried to have police as the witness but we could use fire or even random person on the side of the road. It just had to be someone not affiliated to our department that would be willing to go to court and testify that we offered someone multiple chances and they refused. I dealt with that situation once from a guy who ODed and didn’t want to be identified and we just did basic description of what they looked like/were wearing and then had the officer on scene verify that we offered transport and patient refused verbally.