I couldn't think of a better subject for this, but here's basically what I am trying to do: Machine A is downloading a file from a source that Machine B cannot access. I would then like Machine B to start downloading the file from Machine A while Machine A is still downloading it from a different source.

For this problem, can assume that Machine A is downloading the file in normal sequence. That is, Machine A simply keeps appending data to the file. Also, if the total final size of the file is unknown, can assume that Machine B can be stopped manually when the user knows that the file is done. A key point is to not think of this as trying to get a file from the other source to Machine B as quickly as possible, but rather as getting the file from the other source to Machine A as quick as possible and getting the file from Machine A to Machine B as soon as possible. That is to say:
1.) Machine A can start downloading the file an arbitrary amount of time before Machine B starts downloading.
2.) Machine A should continue downloading from the other source regardless of whether Machine B is downloading from Machine A.
3.) If the transfer between Machine A and Machine B stops, it can be resumed.

So far, everything I've tried only transfers the partial file up to the point where Machine A had already downloaded when the transfer from A to B was started.

Not sure what you want to accomplish. Maybe p2p programs are good direction. BitTorrent for example.

BitTorrent would definetely not work for my purposes, as the other source could theoretically be transferring using any means, not specifically BT. Here's an example to illustrate: Suppose Machine A has another NIC (not the one used to connect to Machine B) which is connected to another network. I basically want to download a file from somewhere on that network to both Machine A and Machine B. A proxy server on Machine A would not work for this, as the intent is basically to get the file on both systems. Furthermore, I have reasons for not wanting/not being able to allow Machine B any access to that private network, so Machine B has to get the file from Machine A. The current solution is to wait for Machine A to finish downloading, then start the transfer to Machine B. What I want to do is start the transfer from Machine A to Machine B while Machine A is still downloading so Machine B does not have to wait.