Taken from the official support website:
Profile improvements and changes
Latest console software picture The latest console software includes
the following profile improvements and changes:
You can download your Xbox Live profile to multiple consoles. However,
you can only sign in to one profile at a time. Your profile is
automatically updated on your console and Xbox Live, whenever you sign
in or out of your profile. To help keep your account safe, you can
choose to require or not require your Microsoft account password on
any console, For more information, see Password-protect your Xbox Live
profile. If the last time you signed in to your profile you were using
a different console, you’ll see a notification of this when you sign
in. For more information, see Your gamertag was last signed in on
another console. It takes significantly less time to download your
profile (called "recover" in previous versions of the console
software).
https://support.xbox.com/en-US/my-account/manage-gamertag-and-profile/profile-storage-changes
Since this is a three year old post, with enough time the problem fixed itself and now have it saved on every console just only signed in at one time.
You can have two different Xbox LIVE accounts on the same Xbox 360 for sure. I've had 4-5 on mine at one time between roommates and such.
You can also recover your Xbox LIVE Gamertag to any console at any point. Your account can only be "active and updated" on a SINGLE console at a time. (What this means is that it must be recovered to a console before you can use online services on it. This makes it so more than one person can't access your account at once.)
The way that purchasing marketplace items works is this:
Whenever you buy any item on the marketplace, it is tied to both your gamertag and your console. Let me give you an example.
Your gamertag is PersonA and your friend's gamertag is PersonB. Let's also say you have a buddy named PersonC who is visiting and has his Xbox LIVE account on a thumb drive.
Your Xbox 360 will be XboxA and each respective person's Xbox will be XboxB and XboxC.
Following along?
You buy an Arcade game on XboxA as PersonA. This means that ANY PERSON (A,B,C,etc.) can play this game on XboxA (YOUR Xbox), OR you can download this game to play it on XboxB or XboxC on your account only.
Make sense?
To summarize--when you purchase the game as PersonB on XboxC (just a random example), either PersonB can play it on any Xbox, or any person can play it on XboxC.
If this doesn't make sense I can clarify further.
Hope it helps! Happy gaming. :)
Edit: I wanted to add a tip. Say you and a buddy both own Halo: Reach and you want some new DLC for 800 MSP, right? Well, if one of you knows you will always be using the game on your own name on your Xbox and no one else will be, you can buy it on his Xbox. That way, anyone on his xbox can play it (him) and you can play it anywhere (on your own Xbox).
It's not illegal or cheating the system or anything, since the permissions are still restricted as such.
Best Answer
No.
To be able to chat in a party, you need to be signed in with an Xbox Live account. On Xbox One, you need to have a Gold subscription to chat in a party. On the Xbox 360, you can only chat one-to-one without a Gold subscription.
In your case, only one account is signed in, therefore only one account is allowed to chat in the party. To be able for both of your boys to chat, the second child would need to sign in with an account that holds a valid Gold subscription.
I would recommend you set up a family using a parent account that holds the Gold subscription. That way you can easily share it with both your children without any additional fee and, on top, you would be able to play around with parental controls.
Here is a link on how to set up the family and related accounts!