Depends massively on the type of game. In general the two techniques are heavily compressing the data format sent over the wire (e.g. using 16 bit floats instead of 32 bit where it doesn't matter, sending 3 element quaternions instead of 4 and recalculating w), and only sending what the player needs to know based on a radius or some other criteria.
Eve Online slows the entire simulation down when huge battles are happening, but they also split simulations across servers based on which system you are in.
Modern WoW shards plays heavily within zones, which is why you will almost never run into the same people while adventuring. They also attempt to limit how many players will be in the same shard at the same time. However there are ways around that through grouping, so when larger battles start happening things get laggy. You press buttons, they are queued up, and can take upwards of a second to show their effect in-game (they must go to the server and back).
Will this change with technology like Google Stadia? Could a massive compute farm calculate all of the state for all players, and then just stream video data down?
My guess is no - you'd want the rendering to happen on the edge close to the player, but with a sufficiently advanced compute topology maybe some tasks can be farmed down the graph?
My understanding is that the client-side isn't generally the bottleneck for MMOs, it's all the server-side computation that has to happen to make it all run. Calculating behaviors, physics, etc for hundreds or thousands of players simultaneously is really hard, and Google Stadia doesn't change that in any meaningful way.
Edit: Well, now that I think about it, it depends entirely on the game; PlanetSide 2 is an example I can think of off the top of my head that brought my computer to its knees during big battles. But most MMOs aren't exactly pushing the envelope of graphics quality or physics simulation, so it's not as big of an issue.
Eve Online slows the entire simulation down when huge battles are happening, but they also split simulations across servers based on which system you are in.
Modern WoW shards plays heavily within zones, which is why you will almost never run into the same people while adventuring. They also attempt to limit how many players will be in the same shard at the same time. However there are ways around that through grouping, so when larger battles start happening things get laggy. You press buttons, they are queued up, and can take upwards of a second to show their effect in-game (they must go to the server and back).