The easiest way is probably to tee the message to stderr as well as stdout:
echo "Script finished on date" | tee /dev/stderr \
| /usr/bin/Mail -s "Script complete" "myaccount@myserver.com"
tee duplicates its input to multiple destinations, including stdout. By default, both stderr and stdout go to the screen; you're redirecting stdout to Mail, leaving just stderr going to the screen.
If you need it in stdout for some reason, you could redirect it back using a subshell (or several other ways):
(
echo "Script finished on date" | tee /dev/stderr \
| /usr/bin/Mail -s "Script complete" "myaccount@myserver.com"
) 2>&1