So I have Skyrim, but part of the disc is scratched and it won't ever get passed the standard Skyrim loading screen when I try to load a saved game. If I was to install it on my Xbox with another disc (a friends) would I be able to play that installation using my disc?
The answer is - it depends.
There's a type of "authentication check" that does a rudimentary "is this a valid Xbox 360 Skyrim disc" that happens when you first start the game. The intent of this check is to ensure you're not sharing or selling the disc after install.
This isn't anywhere as intense as the actual game itself, but there's still a chance that the part of the disc it checks has been damaged. If it hasn't been, then you'd be fine. If it has, then this technique will not work.
I've seen discs go either way though, so the only way to know is to try.
I can answer this to some degree, because I've dealt with the problem before. My grandson had Call Of Duty and he would always play on disk.
I always played on install and disk. He had damaged his disk, so I went over his house and installed the game off of my disk, which resulted in him being able to play with no problems. Therefore, it does not matter if you are using the same disc, as to it continuously referring to the disk after install obviously does not matter, since his disk would not play.
It appears to be the first time that this is happening. It does just a small system check for a disk that is not damaged.
Now, what does that allow? Legally, you cannot make a copied disk and play it. But you could spend pennies on the dollar for certain damaged disks, rent the disk from a kiosk or video store then install and play the game.
In other words, Microsoft tries to prevent most piracy scenarios, but it does not go overboard to prevent them all, and it most likely does not matter if you had purchased an original disk.