How to send content pipeline output to the build dir

I’m a bit confused about the content pipeline output. By default the .xnb’s end up in GameDir/Content/bin.
But the game executable runs in GameDir/bin/Debug and GameDir/bin/Release.
The .xnb’s should find their way into GameDir/bin/Debug/Content and GameDir/bin/Release/Content for the game to run.

But this does not happen on its own in my project. Is it necessary to change settings to make the files copy over to the right folder?

There are .targets files in the current project templates that automate this. Try creating a new project from the templates and add content to it. Building the project will result in the XNB files in the protect output directory. Then compare your .csproj files with that of the new project, particularly at the top and bottom of the file where it references the .targets files.