After setting up a network play at home in my own network when other players start Network Play and connect to the IP address with the port 35000 and the specified  password set it always says: "Need An Account To Connect"

Even if advertising activated the game is being found and when connecting the same message.

What do the other players need to set for the account?

Every time you play the game, you are using and "account", which basically is a player configuration with your record, rank, etc. This account IS NOT external, it is just data within the game files. You can access the account via the Administration tab to set preferences and such. 

So on the remote side, when you want to play a game, you have to set your account up that you are going to use when playing the game. Click on the "Account To Use" button  to select from the accounts on the machine.  There is also a "new" option.

Again, the word account does not mean that anyone is looking or storing your data or you have to register anything. it is just used so that you can play the game under multiple profiles as some people do not like to use the same profile for email as the do solitaire or network. Password can be different, as can other preferences.  All of this is just files in the game data, not shared. Only your "account name" and rank and commander name are sent to the server game I believe. Not your password, not other prefs.

The accounts are set on both computers with different names and different passwords. They are also unique on each computer, no other accounts there. And it’s still saying “ Need An Account To Connect” (s. screenshots attached)

First client with “dietrich” as account and being the server :

Second client with “machro” as the account:

In that last image, notice the label "Account To Use". Click on that button below the label and set it.