Receiving and printing a fax are two different things.
You say that when the phone rings, you pick up, listen for the 'eeeeeeeeeeeeeeeeeeeeeeeeeeeeeeeee' from the other fax and then press # to start the receiving process.
At this stage you should hear your fax going "iiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiiii" and the two faxes start doing the fax thing.
As long as you hear the handshake ("eeee" "iiii") begin you should be OK, unless there is some basic problem with the handshake - like your fax is set up only to receive certain standards of fax. I can't remember the details (because most/all faxes just send/receive all the variations), but if your fax and the other fax can't agree on the setup details then no fax will be sent.
However, if it appears that the fax is being received (the line stays up for 60-90 seconds for example, or you can hear an ongoing "eee" "iii") but not printed then you might need to check that there is toner/paper in the machine.
Are you able to use the photocopy function on the fax ?
z