Hosted Runtime runs on a new machine instance in Google Cloud. You don’t need to set-up any hardware. But you may need to install a few libraries every time you use it.
Local Runtime runs on your machine at home. You need to install Python, Jupyter, and set-up some forwarding. It is useful if you have a lot of data to process locally, or if you have your own powerful GPU to use.
In most cases, I use Hosted Runtime.