You are talking about the Alice and Bob situations.

Alice has a pair of keys. Private and public.

Bob has a pair of keys private and public.

Bob and Alice swap public keys.

Alice uses bobs public key to encrypt a message, when Bob receives the message he uses his private key to decrypt the message.

The private keys that do the decryption are never swapped. Only the public key are swapped which allow encryption.

If some else has bobs public key, they can not decrypt a message because the key they intercepted is used for encryption.

There is also another method called a Diffie-Hellman exchange. Where Bob and allice still swap public keys but Bob uses his private key and allices public key to do a maths equation to get a secret. Allice will use her private key and bobs public key doing the same math equations will result in the same number.

So the secret was never shared between them they both did the same equation and came to the same result.

With asymmetric encryption there are 2 keys - 1 is public (= everyone can look it up) and 1 is private (only the receiver has that key). Those are mathematically related.
When I send a message, I use the receivers Public key to encrypt the message - so that message is only decryptable with the private key, so the recipient alone can decrypt it.

There are a few ways you can send encrypted information, but they all boil down to the same concept: Create a problem that is very difficult to solve unless you already know part of the answer. For example, I have a number that when divided by 7(a prime key in this case) has a remainder of 3. It could be 10, 17, 24, etc.... We then take that number's remainder, raise it to a power and multipy it by a large number to create a huge result. I send this result to the other party without sending my prime, along with the number I multiplied. They choose their own prime and do some more math with my result and send back the result. Since I know my prime, its simple for me to get the result but anyone watching would have to do a lot of math to get the answer.

Two excellent videos on the subject: https://youtu.be/NmM9HA2MQGI https://youtu.be/Yjrfm_oRO0w

At the most basic level, what you need to understand is that each user of a service creates a key pair. A PRIVATE key that DECRYPTS messages and the matching PUBLIC KEY that ENCRYPTS data.

So if I give you my public key, you can use it encode your messages to me.

When data is encrypted with a public key, it undergoes a mathematical transformation. The specific transformation is designed so that the data can't be feasibly reverted (decrypted) using the same public key. Only the private key, due to its unique mathematical properties, can reverse this transformation.

By no means an expert, bit I'll try: One technique would b asymmetric encryption. Every participant has two keys, a public and a private one. When I want to send you an encrypted message, I encrypt the message with your public key. This key you can make available in any way, it can't be used in a harmful way. The message I encrypted with you public, you can decrypt using your private key, and only with that. Like this, you only need to exchange public keys used only for encryption. So no useful information for an attacker. And private keys never need to leave your hands.

https://pi.math.cornell.edu/~mec/2003-2004/cryptography/diffiehellman/diffiehellman.html

To put it simply, one party openly distributes keys that can ONLY ENCRYPT but not decrypt a message. So I can encrypt a message but only the original key distributor can decrypt it. Thats how a first secure communication is established that can then be used to exchange any initial secrets for further communication...

https://youtu.be/AQDCe585Lnc