here's some additional things it would have to do.
1. dns lookup of the other server to resolve the hostname
2. issue the query to the other server (depending on congestion and server load times will vary
3. wait for the other server to run the query (select, sort, etc...)
4. wait for all the data to get sent back. depending on how much data is supposed to be returned will take longer
5. and this one is optional but repeat all previous steps if you now need to issue additional queries based on the last result(s).
i really wouldnt recommend using a mysql server on a totally different network. there is a high amount of overhead, plus you know how every once in a while you type in an internet address to your browser and it just times out, then you hit refresh and its fine, you also run the risk of timeouts like that or just flukes in resolving the host.
edit: not to mention you are transmitting potentially sensitive unencrypted database passwords and data of the net that could be sniffed by hackers trying to intercept packets, however that most likely wouldnt happen. not to mention your bandwidth usage may shoot thru the roof on both servers for the addition up/downloading going on.