One of the most entertaining things about this game for me was playing around online with people, be it friends or strangers.
If I were to just buy, download and install the game right now from GOG, completely vanilla (no unofficial patches or anything), would I be able to connect online and play? Does it work? If it does work out of the box, is anyone playing? I'm not interested in playing on Hamachi or through GameRanger or whatever third party services are out there. I just want to know if I can play the game multiplayer normally with this GOG.com release?

There's a thread about this.
In short, to be able to play it online currently you'll need to utilize the LAN functionality over a VPN. This just makes computers across the internet think they're connected together in same room.
Hamachi is software that makes this easy and what most people who still play it use.