How do I share purchased iOS apps without my Mac? Is there a way someone else can share a purchased iOS app with me when I only have my iOS device with me but no Mac but the other person has both his Mac and his iOS device around?

Is there also a way when we both only have our iOS device around?
Yes. Assuming they haven't run out of authorizations, they can sign into the app store on your iOS device and download the apps they wish to "share".

Just go to the preferences app, Store and sign in another Apple ID. 

You'll have that app to run until you erase it or the whole device. It of course won't transfer that app to a computer where your friend hasn't authorized iTunes, but iOS will be happy to run that version of the app indefinitely.