setuid sticky bit

I have a binary. It is having the following permissions

rws rws rwx mqm:mqm runmqtrm

The same program on another machine is

rws rws rwx root: mqm runmqtrm

This program is a setuid program.
This is what my understanding is. Whatever user the program is started under, it will finally be running as rmuser.

Now if I start the program as root on the second one it starts ok.
If I start the program as mqm on the first box it doesn't work. On the second box if I start it as root it works.

Can somebody explain me this.

Thanks