You don’t need WP Engine to develop a site that is hosted somewhere else, however you’ll need to do more manual steps to import the site as well as deploy it when the work is done.
Basically, you can create a backup on the remote site. Most backup plugins will give you what you want which is basically a zip file that contains the wp-content folder along with a database dump.
From there, import the site into Local by dragging and dropping it onto Local.
Deployments are basically just a reverse of that process. You can right-click on a site in Local and select “Export” which will give you a zip of everything you need to deploy. Each host is different in terms of what they need to deploy, so be sure to search here in the Local forums, or even on your hosting provider’s documentation for the preferred way of deploying a site. Good luck!