Does CloudFlare act as a standalone cloud server, or does it only cache files that already exist on your server? Is MaxCDN the same?
"or does it only cache files that already exist on your server?"
We don't host content. We will cache static content files on your site by default.
According to Cloudflare TOS
SECTION 10: LIMITATION ON NON-HTML CACHING
You acknowledge that CloudFlare's Service is offered as a platform to cache and serve web pages and websites and is not offered for other purposes, such as remote storage...
MaxCDN has Push Zones WHAT IS A PUSH ZONE? A Push Zone is a zone where the user uploads files directly, like you would with a regular hosting account. The zone is then synced out to edge locations. Push Zones are used for distributing files larger than 10MB, or for files that rarely change (such as PDF’s, Word documents, image galleries, etc.)
Before assuming you can store files on either service providers servers. Why not ask them directly?