They exchanged letters, debated theology, sent gifts. Saladin sent his personal physician when Richard was unwell. Meanwhile Richard tried several times to meet up and hang out with Saladin.
How common would this type of enemy commander relationship be at the time? Was it common knowledge; if so, how did the others on either side feel about it?
The idea that they had an "extremely good relationship" is a bit overblown I would say, and since it's mostly a modern invention, contemporaries don't really say anything about it. I don't know if I can talk about how common it was in general, but I've written a bit about Saladin's relationship with Richard and other kings in some previous answers:
What did Saladin think of Richard the Lionheart and Baldwin IV?