No need for you or anyone you're playing with to own Half-Life proper.
That means you can head to Steam, download the game, and start playing co-op Half-Life right now. The seventeen-year-old Half-Life mod is available as a free standalone game, and-thanks to a fair bit of generosity on Valve's part-includes all of the Half-Life campaign maps. Here's a bit of nostalgia for you: Sven Co-op is now on Steam.