As very correctly noticed in the comments, there is a hasty, common, still unforgivable mistake in my original formula.
The correct formula is
=QUERY(Responsibilities!A2:C,"select B, C where month(A)="& month(A2)-1&"")
Please do notice that in order to get the current month one needs to subtract 1 month.
Because in a query formula, when expressing months in numbers
January corresponds to
So the order of the months is:
You would need 2 formulas in your dashboard sheet
You would first need to put today's date in a cell and refer to that cell to get the current month. That formula is:
You can now use the following formula to get your results
=QUERY(Responsibilities!A2:C,"select B, C where month(A)="& month(A2)&"")