Is it possible to limit certain users from seeing some cards on the board or rather is it possible to make it so that users see only specifically labeled cards?
If not, is there an easy way to create a "duplicate board" where changes to some cards affect both boards? So we could create one board with all the cards, and a duplicate with only cards with a specific label. So that someone wouldn't have to make sure both boards are synced.
Right now, card specific permissions are not available for boards in Trello. Once a user has access to a board, they will be able to see all cards on it.
Your best bet is indeed to create different boards with the cards required. You can copy specific cards, lists or the entire board and then remove any unwanted cards: http://help.trello.com/article/802-copying-cards-lists-or-boards
Also becoming quite the dealbreaker. Unito seems to be the only way to do something like this. Basically our solution is to have Unito automatically sync all cards from our executive-access board with a certain label to a separate board and back. Then that secondary board will only see the cards with the label, for their project.
Added benefit over the manual process is that when something is changed it reflects back on the executive-access board.
Major downside right now though is that Unito doesn't seem to be able to sync checklists within cards, as well as any traits which are power-ups.