There appears to be a bug recently introduced that causes the shipping amount not to be displayed. The shipping amount is included in the total price, it's just not being displayed in Checkout in all instances.
Background:
We have a manual shipping method for "local pickup" with no cost associated. When selected the shipping amount "0.00" is displayed as expected.
We also have calculated shipping methods for UPS, USPS, and FedEx. When the shipping method chosen is anything other than zero, the shipping amount isn't displayed. I've confirmed it's not a CSS issue, checking in Firebug, the amount isn't even being output.
Once discovered, I went to an older / development installation that displayed shipping amounts properly, upgraded it to 2.2.0, refreshed the checkout and the shipping was no longer being displayed.
Shipping display appears to be working as expected on your demo site, so there may be some other factors to recreate.