This all depends on your hosting policy. Most hosts block any attempt to access mySQL form anywhere but localhost - this will stop you connecting directly from your pc to your hosted service.
A solution to this depends on how up-to-date you want this synchronisation to be and how complex you want your solution. If you need to replicate once a day then one thing you could try is to use a cron job on the web server to dump the data to a file in an FTP directory and then use PHP's ftp capability to retrieve the file and run it through mysql. Be sure to be VERY careful to have everything nice a secure, though.
If the database is HUGE, then you will need to be even smarter and only dump data that has changed since the last run.
I regularly (but not frequently) take data off one of my hosts and load it up lcoally to investigate client issues. However, I tend to do this manually using PHPmyadmin on both machines. The dump facility creates the sql statements to replicate the data I need. It's very simple and suits my needs.
