If we split apart your command apart with the result of each command after the #
echo "foo" >&2 # echo "foo" and redirect to fd 2 (/dev/sdterr)
| #pipe stdout to
tee /dev/tty #both send stdout to file /dev/tty, which is terminal file that can output both stdout and stderr depending on what you pass to it (so you probably want /dev/stdout/ or /dev/stderr directly instead) and pass it along to the next pipe
| #pipe stdout to
logger -it "my_script"
So it depends on what you want to do (In the above foo gets redirected to stderr and nothing gets piped to tee)
If you want to print foo to stderr and pass stdout to your script you can just do
echo "foo" | tee /dev/stderr | yourscirpt
Then tee will print to stderr and foo will get piped as stdout to yourscript.