Try not using encryption (if you can) as a first step of debug.
Something in the fashion of FTP or simple netcat [nc] utility from both sides (one listening other sending)
Try an NFS mount if you can, and compare speeds.

Couple of questions :
Are other clients (the servers you are copying to) exhibiting the same problem or just that one ?

If all the clients you are copying to experience slow copy -->
Check the netstat command (-s -i) and look for drops, errors and re-transmissions on the server side (the machine you are copying from).

How are disk response times and queues ? Are disks on the server side saturated perhaps ?
If disk read can not give you more 5MB/s no network will achieve more (unlikely, 5mb/s is quite low for todays 'standards', but check, iostat will be your friend here).

If only one (or couple of perhaps in same network VLAN), check out the clients with above commands.

Take output of commands before and after the action and compare.
The before is important, cause those counters are not restarted since boot probably, as a baseline.

Is the copy done over internet or intranet, are firewalls and/or packet inspection/throttling active on those networks ?

This is where netcat may can come into play, for instance, you can throttle 'per app' in firewall like ssh (port 22), nfs (port 2049) but netcat can use any port, test a couple on server/client, try using high port numbers.

And yes, i love netcat