What you are doing is exporting your key. Your public key is indeed something you can (and should) share as it enables others to verify that you are indeed who you claim to be (or more accurately, that you're in control of the private key that's linked to that public key). So while you should share your public key, your private key must remain private.

What these people on the dark web are doing is one step further: they sign their messages with their private key. This creates a cryptographic signature that's different for each message (changing a single character in the message will generate a wildly different signature). Anyone with the public key can simply copy that message including the signature and validate it. If even a single character of the message was changed, the signature will not be valid. Thus ensuring others that the person who posted the message is indeed in control of the private key.

Signing is different from encrypting: while encryption renders your message totally unreadable to anyone without the correct key, signing doesn't change the message itself. It simply appends a signature allowing others to check that the message wasn't tampered with.

The short answer is no. A bit longer of an answer is that with the public key, anybody can encrypt data. Only the owner(s) of the private key can decrypt the data. That is a key point: encrypted data by itself is meaningless. If you were to attempt to decrypt random data (or change one single character of valid encrypted data), you’d get literal garbage output. But, valid encrypted data and the corresponding private key can always unencrypt back into the original format.

This is why emphasis is always made to never share or expose your private key. Couple the private key with the always-available public key and you’ve got a man-in-the-middle (MitM) attack. This is where an attacker could decrypt the data with the private key, change it, re-encrypt it with the public key, and send it along to the destination without anybody knowing it was altered.

I hope this helps.

I am probably at a similar experience level to OP and have wondered the following: is there a commonly used, or agreed upon repository for identity verification with PGP or similar? It would be a useful thing to use, the problem is that if you're posting something to a public space, not everyone may have access to your public key for verification. Including the key in the message doesn't seem like it would help much, since someone else could just generate two new keys and still claim to be you without any affiliation. Am I wrong about this? Thanks.

EDIT: changed encryption / decryption to hashing / hash-verification. Thanks for the corrections

Not an expert, those who know more please correct me.

From what I understand, what they post is not a PGP key, but the same content published in clear text hashed with their private key. That way anyone can verify that hash with the author's public key to check it has been generated with the private one (that only one person should have).