Any data you wish to share among all peers must be communicated somehow, either via the built in sync or by messages.
Once a peer receives the data, it is up to the local client how to store and display it.
In the case of an MMO, depending on your scope, you might want to keep in mind the frequency and size of broadcasted updates as to not overload the host's bandwidth (remember that bandwidth loads scale exponentially with number of players).
You can use global variables and layers to persist across layouts. Some objects are naturally global like array, others like sprites you can add the persist behavior to.
This is when utilizing the built in multiplayer plugin. If you are using an external backend service, you have many different options available depending on the chosen provider. If you are trying to make an MMO, I would definitely suggest a backend service as optimal.