I can use SP.RequestExecutor.js to retrieve typical SharePoint elements like lists and list items from the host web but I get an Unauthorized response if I try to talk to anything within the hostweb/_layouts/15/_api_bin/ space. I have added the host web's endpoint to my AppManifest.
I tried using the WebProxy method described here (https://msdn.microsoft.com/en-us/library/office/fp179895.aspx) but it still returns an unauthorized. I have no problem accessing the services (using the same user) if I am in the host web itself or making the call from a page hosted on the host web domain.
If anyone has a code example or tip they can share, I'm sure a lot of other people would find it very helpful. I've seen other requests for help in trying to talk to, say, the Excel services on the host web but I haven't seen any solutions at all.