Ssh – Copy files without encryption (ssh) in local network

For a single file instead of using SFTP you could pipe the file over ssh using cat or pv at the sending side and using tee on the middle server to both send the data to a file there and send a copy over the another ssh link the other side of which just writes the data to a file. The exact voodoo required I'll leave as an exercise for the reader, as I've not got time to play right now (sorry). This method would only work if the second destination is publicly accessible via SSH which may not be the case as you describe it as a client machine.

Another approach, which is less "run and wait" but may otherwise be easier, it to use rsync between the server and client B. The first time you run this it may get a partial copy of the data, but you can just re-run it to get more data afterwards (with one final run once the Client1->Server transfer is complete). This will only work if the server puts the data direct into the right file-name during the SFTP transfer (sometimes you will see the data going into a temporary file which is then renamed once the file is completely transferred - this is done to make the file update more atomic but will render the rsync idea unusable). You could also use rsync for the C1->S transfer instead of scp (if you use the --inplace option to avoid the problem mentioned above) - using rsync would also give you protection against needing to resend everything if the C1->Server connection experiences problems during a large transfer (I tend to use rsync --inplace -a --progress <source> <dest> instead of scp/sftp when rsync is available, for this "transfer resume" behaviour).

To summarise the above, running:

rsync --inplace -a --progress <source> user@server:/<destination_file_or_folder>

on client1 then running

rsync --inplace -a --progress user@server:/<destination_file_or_folder> <destination_on_cli2>

on client2 repeatedly until the first transfer is complete (then running once more to make sure you've got everything). rsync is very good at only transferring the absolute minimum it needs to update a location instead of transferring the whole lot each time. For paranoia you might want to add the --checksum option to the rsync commands (which will take much more CPU time for large files but won't result in significantly more data being transfered unless it is needed) and for speed the --compress option will help if the data you are transferring is not already in a compressed format.

I did not want to leave this as an answer but the comments section is gettinfg larger and you will get notification from SE to carry the conversation to a chat window.

So, hit enter a couple of times and type ~? no spaces. no nothing.

you should see something like this (or exactly like this)

Supported escape sequences:
  ~.  - terminate connection (and any multiplexed sessions)
  ~B  - send a BREAK to the remote system
  ~C  - open a command line
  ~R  - Request rekey (SSH protocol 2 only)
  ~^Z - suspend ssh
  ~#  - list forwarded connections
  ~&  - background ssh (when waiting for connections to terminate)
  ~?  - this message
  ~~  - send the escape character by typing it twice
(Note that escapes are only recognized immediately after newline.)

at this point type ~C and you should see ssh> prompt. At this point type

ssh> !scp /path/to/some/file/on/your/local/machine user@remoteserver

since you are already authenticated, it should transfer the file over the existing connection.