Friend got an S, I didn't receive any chest?

You don't get a chest when someone else gets an s

Had you earned a chest on the champion you were playing before?

E.g. If I get an S+ on Jinx I get a chest that's tied to Jinx. If in the next game my friend gets an S- on Katarina, and I'm playing Jinx, I don't get a chest because I've gotten Jinx's chest already (even though I don't have one for katarina). My friend will get a chest if he has one available on Katarina.

Also make sure you were dynamic queued with them. You only get it if you queued up. If you happened to be match-made into a game with your friend, it won't count.

Did the champion you were playing earn their chest for the season already?

Did you have a chest available to be earned? It shows above the win of the day timer if you do.