Check for force air ducts that might meet somewhere you can get to. I've run cables in apartments many times. You just need to get creative sometimes. A hole in the wall might not be an option that the landlord approves of, but you can get away with a lot if you're careful to conceal it, and of course patch it up if you ever leave.
In one apartment I was at, I found an electrical box that was nearly in the exact space on the other side of the wall, which allowed me to go right next to it and be concealed by the plate cover... now I don't suggest using an electrical box unless you KNOW what you're this can be VERY VERY bad if you hit a live wire...
Other ideas are to add a low voltage box and wall plate on either side of the wall, and if you make it look nice many land lords wouldn't even notice.. (The key is doing it right, so that it matches everything else)
If you have basement access, you might be able to drill a small hole under the carpet, near the corner of each room, pretty much hiding the hole, and it's really easy to pull the carpet back over into a corner without any damage.
I've seen a lot of people do these kinds of things half-assed, and that for sure will raise flags. If you take your time, and do it right it's worth it.
Knowing how terminate cat5 is helpful, since you can use a much smaller hole if need be. Drywall holes are very easily patched.
If there is a cold air return on the wall, they might be shared between both room.
Be creative, be safe, and be smart.