Browers can't feasibly stop web pages from talking to private (local) IP addresses
[...] The major browsers I've tested — Safari, Chrome, Firefox — all allow web pages to send requests not only to localhost but also to any IP address on your Local Area Network! Can you believe that? I'm both astonished and horrified.
(Johnson mostly means things with private IP addresses, which is the only sense of 'on your local and private network' that can be usefully determined.)
This is a tempting and natural viewpoint, but unfortunately this can't be done in practice without breaking things. To understand this, I'll outline a series of approaches and then explain why they fail or cause problems.
To start with, a browser can't refuse to connect to private IP addresses unless the URL was typed in the URL bar because there are plenty of organizations that use private IP addresses for their internal web sites. Their websites may link to each other, load resources from each other, put each other in iframes, and in general do anything you don't want an outside website to do to your local network, and it is far too late to tell everyone that they can't do this all of a sudden.
It's not sufficient for a browser to just block access by explicit IP address, to stop web pages from poking URLs like 'http://192.168.10.10/...'. If you control a domain name, you can make hosts in that have arbitrary IP addresses, including private IP addresses and 127.0.0.1. Some DNS resolvers will screen these out except for 'internal' domains where you've pre-approved them, but a browser can't assume that it's always going to be behind such a DNS resolver.
(Nor can the browser implement such a resolver itself, because it doesn't know what the valid internal domains even are.)
To avoid this sort of DNS injection, let's say that the browser will only accept private IP addresses if they're the result of looking up hosts in top level domains that don't actually exist. If the browser looks up 'nasty.evil.com' and gets a private IP address, it's discarded; the browser only accepts it if it comes from 'good.nosuchtld'. Unfortunately for this idea, various organizations like to put their internal web sites into private subdomains under their normal domain name, like '<host>.corp.us.com' or '<host>.internal.whoever.net'. Among other reasons to do this, this avoids problems when your private top level domain turns into a real top level domain.
So let's use a security zone model. The browser will divide websites and URLs into 'inside' and 'outside' zones, based on what IP address the URL is loaded from (something that the browser necessarily knows at the time it fetches the contents). An 'inside' page or resource may refer to outside things and include outside links, but an outside page or resource cannot do this with inside resources; going outside is a one-way gate. This looks like it will keep internal organizational websites on private IP addresses working, no matter what DNS names they use. (Let's generously assume that the browser manages to get all of this right and there are no tricky cases that slip by.)
Unfortunately this isn't sufficient to keep places like us working. We have a 'split horizon' DNS setup, where the same DNS name resolves to different IP addresses depending on whether you're inside or outside our network perimeter, and we also have a number of public websites that actually live in private IP address space but that are NAT'd to public IPs by our external firewall. These websites are publicly accessible, get linked to by outside things, and may even have their resources loaded by outside public websites, but if you're inside our network perimeter and you look up their name, you get a private IP address and you have to use this IP address to talk to them. This is exactly an 'outside' host referring to an 'inside' resource, which would be blocked by the security zone model.
If browsers were starting from scratch today, there would probably be a lot of things done differently (hopefully more securely). But they aren't, and so we're pretty much stuck with this situation.