Ok
Is Chess.com Allowed to Read Our Messages

Surely they must have the ability. If someone's abusing another member via private messages, and the abused member files a complaint how else could chess.com corroborate the claim of abuse in order to take appropriate action against the abuser?

Straight from the TOS (Legal, at the bottom of the page)
You acknowledge, consent and agree that Chess.com may access, preserve and disclose your account information and Content if required to do so by law or in a good faith belief that such access preservation or disclosure is reasonably necessary to: (a) comply with legal process; (b) enforce the TOS; (c) respond to claims that any Content violates the rights of third parties; (d) respond to your requests for customer service; or (e) protect the rights, property or personal safety of Chess.com, its users and the public.

Surely they must have the ability. If someone's abusing another member via private messages, and the abused member files a complaint how else could chess.com corroborate the claim of abuse in order to take appropriate action against the abuser?
Either a screenshot or an inspection of the offending message with the consent of its receipient. What the child is talking about is serreptitious reading of messages. Which I am sure goes on.

So can they read other's people messages, and then tell someone else about them just because they want to know?

chessdex, what the hell are you talking about? Admins don't go around reading random peoples messages let alone telling others about members about the content of the messages for no reason. Don't you think that they have better things to do with their time?
Honestly...

I'm pretty sure they have done this before just to tell other people about another people's secrets. I think, I'm not sure, but if it did happen I don't think it would be fair

I'd be pretty surprised if that was happening, but if they suspect cheating, abuse or that a user is a returning banned user, for example, I wouldn't be surprised if they go in to have a look at their messages to see if they could confirm. Sharing the contents with other members, though? Surely not.

I have administered mail boxes for years and me nor my colleagues had even the idea to check emails from people. Not only as some code of honor, but also simply the fact that you have better things to do like Kim Young Un said and it was made difficult to do - not impossible though.
In my case of Lotus Notes was it made difficult by restricting the access rights of normal admins like me and all my colleagues.
A super admin account could manage and read all emaildatabases of clients, but we as admin were never allowed to use that account. If needed, then could we access that account and login with it, but in daily life, we weren't allowed and able to use it.

I have administered mail boxes for years...
...Lotus Notes...
So sorry to hear this.

@chessdex: why? Please explain. AFAIK do he and I agree this time.
@TheGrobe: lol, it was a very good platform in that time. Years ahead of other mailing programs. Don't know anymore since the start of version 7. Please don't tell me that you have worked with Exchange and think that it was/is a better program?
If they are, at what times, or just whenever they like? Also, can they tell other people about someone else's messages? Just curious