You're not dreaming – an official suite of replay features is, in fact, entering the Public Beta Environment. For a limited time post-match, your performances will be stored and available for viewing on Riot's servers with any replays you've deemed fit enough to view being filed safely away in your summoner's profile. From there, you're free to share the replay files with friends. Happily, the replay UI will be nearly identical to the spectator controls you're used to.
So, when will these features hit the live game? Well, it'll be a while. Status Kwoh explains in his board post:
“We’ve been working hard on this much-requested feature for a while now, but there’s still work to be done before it hits live. Rolling out replays to millions of players is no small feat, and there are some pretty big infrastructure and service challenges that we still need to tackle before they’re ready for a live release.
“So why the long testing period? Well, on top of the service challenges, there are some fun complications involved with backwards compatibility. As such, we’d love replays to get the same thorough, extensive testing as Spectator Mode. As you recall, we held numerous beta periods over the course of that launch, and these periods of testing really helped us fine tune that feature.“
The call to (testing) arms has been issued. If you've been itching for a way to prove your penta-kills to your dubious buddies, now's your chance to vet your means. You can test these features out on the PBE.