Sharing SSH keys
Update: as an additional security recommendation, you should generate a new set of keys for a new machine and send your new public key out to the various hosts you use it on, rather than copying your private keys. If you're just moving everything to a new computer however, you can take your keys with you, but remember to destroy them securely on the old computer.
The correct answer is to copy your .ssh directory from the old machine to the new. This part is easy (scp -r .ssh user@newmachinehost:~ will do fine—or you can type the keys in character-by-character, up to you).
BUT—I think the missing link to answer this question is what you have to do after you copy your private keys to the new machine.
I had to run the following for each key (I have 3 separate keys for various organizations)
ssh-add .ssh/[key-filename]
If the filename argument is omitted, id_rsa is assumed.
Once you do this to each key (and enter they key's passphrase if required; it will prompt you), ssh will be able to use those keys to authenticate.
Otherwise, no amount of copying will do much. SSH will ignore the keys in .ssh until they are explicitly used (via ssh -i [keyfilename] ...).
This should work, and both machines should be able to maintain a connection at the same time - I've had to copy my ~/.ssh directory a few times before when hard drives have crashed.
Copying ~/.ssh between systems is fine so long as it's limited to just files like authorized_keys, config, and known_hosts. If you want two hosts to be able to access each other, each host needs its own private SSH key, which must then be added to the other host's authorized_keys file.
It is not a good idea to copy private keys across systems!
Think of real world secrets. Each person who learns the secret increases the chance of it being revealed.
Every time you copy your private key to a new system, you increase your risk of exposure because copied private keys are less secure than the weakest system they live on (because the other systems aren't invulnerable either).
If your laptop gets stolen, you need to revoke all private keys (and saved passwords) that were stored there. This becomes problematic when the only way to log into servers is with that very key. You'd better remember to generate a new key on your desktop and install it on each system you revoke the old key from!
Now consider your desktop gets hacked and somebody steals your private key without your knowledge. Perhaps you had shared this key between your work laptop and your personal desktop, but your desktop doesn't really need access to your work system (because you have good work/life balance). That attacker can now access your work system even without having compromised your laptop. The infosec team at work forces you to hand over your laptop so they can audit it, but after a week of analysis, they find nothing. Not so fun.
These may seem far-fetched and unlikely, especially if you're not a prime target (e.g. an executive or sysadmin), but it's just a good practice, especially given how easy it is to create new keys for each system and install their public keys on each appropriate server. (Consider one of the myriads of config/dotfile propagation systems if this seems daunting.)
Additionally, it means you'll upgrade the security of each key to meet the standards as they improve. As you retire old systems and remove their keys, you rotate out their weaker keys. (And if some trash picker finds your old laptop and undeletes your keys, they won't grant any access.)