No. You should never share your PRIVATE key with anyone. What you do is exchange PUBLIC keys in such a way as to be certain that the public key came from the person in question. Usually email will work, although you could put the key on a floppy disk and trade them in person, if you wanted to be all cloak-and-dagger about it.
Then, when you want to encrypt a message to be sent to that person, you encrypt it with their PUBLIC key. The only thing that can decrypt the message will be their PRIVATE key. That’s the beauty of Asymmetric Encryption; You have two keys, and what is done with one key can only be undone by the other key.
So as long as you keep your private key secure, anyone with your PUBLIC key can verify that a message signed by your PRIVATE key is legit, since the PUBLIC key is the only thing that can verify data that has been signed or encrypted by the PRIVATE key. On the flip side, encrypting a file with someone’s PUBLIC key will ensure that only the holder of the PRIVATE key can decrypt it.
I should also mention the difference between encrypting and signing. Encrypting a message makes it unreadable to everyone except whoever has the mate to the key that did the encrypting. Signing, on the other hand, leaves the message intact and plainly visible to all, and simply appends a signature to the end of the message. Like encryption, this signature can only be verified by the mate of the key that did the signing.