Registering and getting a Foreigner's Residence Permit is your responsibility.
You may let the school, or someone do it for you but it they screw it up, or don't do it, you bear the consequence. First school I worked at, I was in my room and answered a knock on the door. Two cops informed me I was going to be deported. It took a little while to get the story but the short version is, the guy that was supposed to registration didn't do it on time. At the second school, I was told to go to the office. When I got there, the three cops told me to sign a "confession." Why? Because you did not register your residence on time. Again, the lowest guy on the Chinese food chain had screwed up. I protested until I was told "Sign it, it is easier for everybody."
Your Passport, your visa, your work permit, and your residence registration IS ALWAYS YOUR RESPONSIBILITY. Usually the school does it to save you the trouble but if the school screws it up, you suffer. Pointing at a Chinese person gets you squat. TIC
Since 2003, I have done it myself. It takes only a few minutes, the people are friendly, helpful and usually do most of it, filling out the form, for you. Every place I have done registration at has had something in English on the wall telling you this is where it is done. Best part is no knocks on thew door.