Some Blackmagic Design Decklink capture cards support Input and Output but not both simultaneously.
The exception to this is newer "4K" Decklink cards which support Input and Output at the same time as described in the Blackmagic forums:
You can also use a multiple input card such as the Decklink Duo or Quad series if you will be using the card to capture video and also to display the Logosys output.
For example, the following cards support the following:
Decklink Duo: 2 Channels, 1 Input and 1 Output simultaneously or 2 Inputs 0 Outputs.
Decklink Duo 2: 4 Channels, Any combination of four. (3 in 1 out, 2 in 2 out, 4 in 0 out)
Decklink Quad: 4 channels. Any combination of four. (3 in 1 out, 2 in 2 out, 4 in 0 out)
Decklink Quad 2: 8 channels. Any combination of eight.
Contact Blackmagic Design for more information.