I don't see any limitations in the item description, and I wasn't having much luck seeing this referenced elsewhere.

Example: Character A is in the ring, which Character B is currently attuned to. If character B dies, does the ring contain both characters who can communicate with the next wielder?

EDIT: SOLVED

So if someone picks up the ring with a soul in it, they can't attune?

You can attune to a Ring of Mind Shielding that has a soul in it, but you can't add a soul after the first one:

If you die while wearing the ring, your soul enters it, unless it already houses a soul

That means that only one soul can be inside the ring at a given time.

Quote from Weatherguy >>

I don't see any limitations in the item description, and I wasn't having much luck seeing this referenced elsewhere.

Example: Character A is in the ring, which Character B is currently attuned to. If character B dies, does the ring contain both characters who can communicate with the next wielder?

 There's an important line at the start of the third paragraph:

If you die while wearing the ring, your soul enters it, unless it already houses a soul.

As for the attunement criteria, from the rules on attunement:

A creature's attunement to an item ends if the creature no longer satisfies the prerequisites for attunement, if the item has been more than 100 feet away for at least 24 hours, if the creature dies, or if another creature attunes to the item. 

So dying and getting transfered into the ring ends your attunement, and even if it didn't, another creature attuning to it ends your attunement.

Effectively, once you're in the ring no one else can get put in unless your soul gets removed first.

Oof. I totally misread the item. Cheers.

I'm not a DM, but this would be a great way to have side quests in any campaign by releasing a prior soul. Having the party help the soul move on, so it is free to be used by the wearers soul if they die.