Hi cklunar,
I am running Mavericks 10.9.1 on a iMac 2013. The second display is a iMac 2010 in target display mode.
I have Automatically rearrange Spaces unchecked
Ok, I have to say that while I was preparing the answer I played with the two monitors even more and this is now what I understand:
- suppose I have 3 desktops on each monitor: Display 1 (Desktop 1-3), Display 2 (Desktop 4-6).
- I am working on Display 1/Desktop 1, and the app is also on Display 1/Desktop 1 and I right click on the dock icon then I won't see the option of assigning to a desktop, only to a display (see screenshot). For instance I am typing in Chrome, which is in Display 1/Desktop 1. The same would be true if I was in Display 2/Desktop 4 while Display 1 is on Desktop 1. This is why I was getting it wrong.

- I am working on Display 1/Desktop 2, and the app is still on Display 1/Desktop 1 and I right click I will now see the option of assigning to a desktop (see screenshot). The same would be true if I was in Display 2/Desktop 4 while Display 1 is on Desktop 2.

- Some things are odd. For instance, I have iTunes on Display 2/desktop 6 but I would only have the choice of assigning it to Display 2/ Desktop 5 (see screenshot)

I hope it makes sense, and I hope it is useful to others. I am relatively new to OS X.
Mario