Multiplayer question

So I made a single player puzzle game and I enjoyed allot making it. My next big dream is to make a multiplayer game out of it.

I see allot of code what to implement to get the server etc going but I never see someone talk about this is how a multiplayer game is made.

Can someone just explain how such a game is usually developed ?
Like do you make a player 1 en 2 variable and keep the score in a table for each player like this

Player 1 = {score =0,name =nil,andso on}

For player 2 too and try to add to the score in function that is called by 1 and goes to 2

Or make a function that takes a parameter of which player called it.

Do you make double grids for the playing field or just one ?

Can someone please tell me in plain English how such games are coded ?