Others have touched on this, but I will attempt to summarize.....

Each user on each system needs to have a private ssh key, and a public ssh key.

For maximum security, each key pair should be unique.  This means you will need to copy all of the public keys to all of the other servers, and add them to authorized_keys.

For maximum convenience (which equals less security), you can copy the public *and* private key for one system to all of the other systems.  Be careful that the modes on the private keys are set correctly.  And, as has already been pointed out, be aware that with this set-up, if someone breaks into one of the accounts, they have access to all of them with no further effort.