you dont _need_ to have 2 servers, no, the unidentified error means that there is a problem with the settings for the testing server, ussually the Web URL or the Host Root setting. you can use the remote server as both remote and testing server if you like, or you can use a local testing server, that is decision that is up to you.
either way, the settings need to be set correctly for connecting to the server you use for testing.
when the testing server settings are right, you should be able to edit the existing connection and change the Host, Username, Password, and database name for the connection.
the access denied error in your screen shot is as strait forward an error as they come.
it means the MySQL host is not accepting the username and password you entered.
