I would be cautious about having two processors controlling the same SD card, because one may "cache" data (like the disk directory) and then be surprised if the other one changes the disk contents. It could be done if you totally closed the card before accessing from the other processor (but then you would have to negotiate which processor "owns" the SD card at one time). To have them both accessing the card willy-nilly is almost certainly going to lead to grief.
You might find I2C simpler to communicate between processors if you are already tying up SPI with the SD card. You could use SPI for both, but again you would need to decide, at a given moment, if the SD card is going to be active or the other processor.
So you will have to develop a protocol to decide who has access, for how long, etc. and come up with similar buffering for the SCK, MOSI, MISO, and CS lines to share who is controlling them.
CrossRoads is right, and that part will be the tricky bit.