Installation directions don't really differ remotely vs locally. You are still required to have:
A web server with PHP installed
A MySQL server
The OST files from the distribution
If you wanted to run it locally would need to install and configure all of that to get it to work. A basic check list would look something like:
install web server
configure web server to use PHP
setup web page in web server / put OST files in proper place
install MySQL and configure it
ensure that mysql-php and mysql-imap are enabled
create MySQL database for OST
create MySQL username/password and grant permissions to database for OST
run the OSTicket installer