Disclosure: I'm on the Flutter team.
We capture a small amount of analytics from the developer tool itself, primarily to help us spot and fix crashes as well as to help us improve the tooling. Completely anonymous, we prompt you on installation, and you can disable it completely using the command 'flutter config --no-analytics'.
Flutter does not capture or upload any information from users running apps compiled with it. The SDK is all open source, of course, so you can see exactly what we do upload.