Haha. I've actually discussed this very issue with my co-workers as a thought experiment.

The simplest method would be to store the entire replay data on a central server (or cloud) somewhere and allow people to download it while the match is in progress. The biggest drawback to this system is it would require large amounts of storage capacity, even if you're deleting the replay data shortly after the game ends. However, mass data storage is a relatively simple process these days, so this shouldn't prove to be terribly prohibitive for a company like Riot.

If mass storage is a problem, they could go with a P2P system where each player stores their version of events locally, with some "collation" process occurring whenever someone wants a copy of the replay. However, this P2P system has some major drawbacks. First, if one or more players disconnect during the game, there's no way to recover their data. This isn't necessarily a fatal problem, but it does effect replay quality. Second, network bandwidth become the limiting factor with this system. This system can work if the number of people requesting a replay remains low at any given time, but if you've got hundreds (or thousands) of machines requesting your data while you're playing, your playing experience is quickly going to deteriorate. (Imagine you're running Bittorrent in the background while you're trying to play the game, and you'll get some idea of the network load.)

Honestly, I don't know how Riot could decide to go with a P2P system. It's far more complicated to write, is less reliable, and is severely bandwidth-limited. By contrast, mass storage is a pretty much a solved problem these days, which makes the first solution much more practical.

(As a side note: I think Riot is already using mass storage for spectator mode. The replay that you're watching is delayed by three minutes and it doesn't take three minutes for you to join a game, so that data is obviously being buffered somewhere. As I mentioned earlier, it's unlikely that the data is buffered on the players' machines because of bandwidth concerns, so they probably already have a replay server set up to store this data centrally. Of course, they only have to store the last three minutes of play on the server, so they're currently using a lot less space than they would need to store an entire game.)