Just like J Bruni said, it’s a matter of copying the key to the other computer that wants to use it.
For the problem you mentioned about copying the key to another Mac and asking for password, I recommend that you create a private key manually, as described in https://help.github.com/articles/generating-ssh-keys and do not manually add a password at the time it asks for password. If it doesn’t work, create the private key on a Linux or Windows with Cywin and copy the generated key to all computers.
My only fear when using Dropbox to stay with private key sharing is that it might be compromised. Remember that private key is something serious. It can be used to login without a password, which means that if it is exposed, it can cause a lot of damage. Since it’s not something that changes frequently, it’s better to have it in every location.
My solution was to create a new key following Emerson’s recommendation. I tested it on a virtual machine running OS X 10.9 and it accepted the key normally. Maybe the problem is related to key creation
– Yan Minari