No. That wouldn't work because the second "command1" is not the same instance as the first one.
I think the solution here is to use fifos.
Something like that (probably inside a third "controler" script):
command1 </tmp/from2to1 >/tmp/from1to2 &
command2 </tmp/from1to2 >/tmp/from2to1
rm -f /tmp/from1to2 /tmp/from2to1