It depends on the setup; if you're running go-ipfs, the node is a program that runs on your computer and becomes available in your browser, either through its I

It depends on the setup; if you're running go-ipfs, the node is a program that runs on your computer and becomes available in your browser, either through its IP address (yours, optionally public with proper network setup) or through the ipfs.io domain, which access the entire network and can pull from any node that's connected, which includes your own. That's just the Go implementation though, there's an implementation entirely in JavaScript, and it would be more than plausible to bake the service into a browser bundle so it works like you describe. However, I believe there's strength in separation personally. The acts of hosting an IPFS gateway, and accessing IPFS content through a gateway, are separate.