Although perhaps more constraining, why not use a website that uses WebRTC data channels to transfer the files? Then you can be more sure the data isn't persisting in a datacenter somewhere. Plus, it is more plausible that the service can remain free and private.
As you pointed out, it is a bit more constraining due to the support for WebRTC and users behind an SNAT, but I think for the majority of users it works well.