I am just starting to research online multiplayer game development and I have a high-level architectural question regarding how online multiple games function.

I have server-side and client-side programming experience, and I understand how AJAX-esque transfer protocol operates.

What I don't understand yet is how online multiple fits into all of that. For example, an online Tetris multiplayer game. Would both players have the entire Tetris game built out on their client-side and then get pushed "moves" from the other player via some AJAX-esque mechanism, in which case each client would have to be constantly listening via JavaScript for inbound "moves" and update the client appropriately?

Or would each client build out the aesthetics and run a virtual server per game to which each client connects and thus pull and push commands in real-time via something like web sockets?

I apologize if this question is too high-level and general, but I couldn't find anything online that offered this high-level of a perspective on the topic.