Well, for example, you could use Go on the server and Dart in the browser. (I quite like using Go with App Engine.)

Of course that means you can't share code between client and server, but not everyone needs that.

Both Dart and Go can be run in the browser or on the server, so you can share code between client and server regardless of which language you choose, so long as you don't mix them.

Well, maybe. I have not kept up with GopherJS; how are they doing on code size? And while you can certainly run Dart on an App Engine Managed VM, there is no free tier like with Go.

Not sure about GopherJS. I'm confused about how App Engine's pricing model relates to whether or not Dart is well-suited to server tasks?

Well, it's relevant to those of us who are using App Engine. Where else can I can I set up a service for a hobby project without a monthly charge if it's barely used?

Your question is kind of nonsensical. The two languages have little, if any, overlap in use cases.

This isn't true; they're both general purpose programming languages with strong static type systems and decent async I/O stories. You can write a web app or a server in either language, though Dart has a better ecosystem for frontend development, and Go has a better ecosystem for backend development.

Is there any "ecosystem for frontend development" for Go at all?

Dart is dynamically typed.

It's technically optionally typed, but it has a static type checker which puts it more in the "static" camp in my mind. https://www.dartlang.org/articles/language/optional-types

Dart and Go each serve different platforms.