When uploading a png file the thumbnail image renders as a broken image. If I past the URL to the thumbnail into a new browser tab, I get an error saying "You don't have permission to access [URL of file]". The resized image is missing from the _resampled and when I check the console logs I see "client denied by server configuration".

This only happens with png files. Jpg files are resized as intended.

Is this a configuration problem or a bug?

Configuration.
Assuming you're probably not using Apache, you need to allow .png files from the assets folder, and that the web user (whoever runs the PHP script) has permission to write to that (and all sub-)folder(s).

If you're familiar with Apache configs you can look at the assets/.htaccess to see how it does things.

I am using Apache, and there's nothing wrong with the permission settings or the .htaccess file for the assets folder and it's subfolders.

However, I noticed that I only have this problem on my local development machine. Seems to work fine in the production environment, so it seems like this is a configuration problem.

Are you running OSX Yosemite on your machine? Because Apple forgot to include imagecreatefrompng().

I am running Yosemite. That explains it!