SSH authentication between my VPS and Bitbucket fail

Did you add the public key to your bitbucket account? It should likely be added as a deploy key, giving it read-only access to your repo(s).

Additionally, all bitbucket URLs are of the form:

[email protected]:<repo_name>

I'm not sure where you're getting the idea to try and authenticate with your email address, but that's not going to work.

To test if your key has been added correctly, you'll need to follow the same pattern as the repo URLs use:

$ ssh [email protected]

Read up on bitbucket's documentation - they lay all of this out for you.

You need to copy the public key ~/.ssh/id_rsa.pub to the destination host. How you do that on bitbucket I don't know.