Most users will like to show cards ONLY where they are either responsible, or where there is a sub-task where they are responsible.
It would be cool
a) to offer a filter that only shows you cards, where a selected user (or the logged in user) is either responsible, or has a subtask where he is responsible
b) to choose in access restrictions that users can only see cards where either they are responsible or there is a sub-task where they are responsible. Otherwise in some teams they are bombarded with stuff they should not take care of or they should not see.