Earlier quoted context omitted.
The same way you share any of your other data across servers. Save it in a DB or Redis instance that all your servers can reach.
At which point you're hitting it on every request and since it'll be a nice, fast, indexed database lookup you might as well just use session tokens instead of a back-assward blacklist. Do the simplest thing that will work.
This is a complex distributed systems topic, and there are applications for both.