This is not how SSH works, here is the standard procedure:
- You generate your private and public key.
- You send your public key, you don't send the private key.
- Once they have add your public key you have access.
If you don't have a private key to go along with the public key then you can't use that public key since you don't have the corresponding private key. This private key needs to be generated by yourself and never shared, since anyone else that has this private key can access the server under your name.
It is likely they don't know how SSH works, they should expect you send them a public key.
As to why this does work the way this does, let me explain...
Take the second image here from an informational page at Amherst College for example, the top part of this image is the server and the bottom part of this image is the client. The server encrypts the messages using the public key and then it can only be decrypted using the private key.
Therefore, for successful communication you need the private key as well. And for secure communication, you don't share that key with anyone as they will be able to read your messages.