Someone close to you, in person
The hardest one, and the one worth rehearsing. Say it early in the conversation rather than working up to it — a long preamble tells them something frightening is coming and they will have started bracing before you finish.
I want to tell you something, and I want to say up front that I'm not asking you for anything today. my friend is looking for a living kidney donor, and I'm helping. Finding out is free and private, and a transplant center decides. I'm helping my friend look. That means I'm telling people, and I'm asking them to find out whether they could be a donor. Not to decide anything. Just to find out. If that's something you'd ever want to look into, there's a page with my friend's story and a form that's the first step. It's free, it's confidential, and my friend doesn't see who fills it in. And if it's not for you, that's completely fine and I'd never know either way. The other thing that genuinely helps is sharing the page — many donors first hear about it from a friend rather than from the person who needs the kidney.