Can I remove successfully migrated files from their local storage on Magento?

After a successful migration of all your images to Cloudinary, you can indeed delete the images from your Magento server. This will only delete the local images and will not affect the ones on your Cloudinary account.
That being said, note that the plugin is designed to store resources on both your Magento server and your Cloudinary account. This way, you can Disable the Cloudinary module and continue to use the images directly from your server whenever you require (e.g. maintenance, etc.).

Note that the above refers to both our Magento 1 and Magento 2 modules.