Xbox Network – Can Xbox One Player Voice Chat or Party with Xbox 360 Player?

You can be in a party while playing split-screen, but unless the second player has an Xbox Live account, they won't be able to participate in any of the voice chat options that Xbox Live offers. You can plug a headset into their controller, but they won't hear the party/game chat audio, and their mic won't function.

If the options to join a party aren't on the same buttons in-game when you're playing split screen, you can always use the Xbox Guide button to access the party menu and start/join a party from that interface.

Note also that you have to have a second controller plugged in/turned on in order for the "Press A to Join" message to appear in most games (MW3 included). There's nothing in MW3 or Xbox Live that prevents you from doing both at the same time.

Have you taken these steps (from MS Support)?

Install a game on your hard drive

1.Insert the game disc into the disc drive.

Note After you insert the disc, the default action is to automatically start the game. If this happens, exit the game and return to the Xbox Dashboard.

2.From the Home hub, select the game in the drive tray, and then press X on your controller.

3.Select Install.

4.A progress indicator appears while your game is being installed on your hard drive.

Note Installing an Xbox 360 game might take 12 minutes.

The disk is still required to play (needs to be inserted into the Xbox360).

Here is another question listing benefits/downsides of installing games.