Who should keep your work permit?

As a practical matter, I really don't think it matters much.  The permit is only for you to work at your place of employment on the permit and nowhere else.  It is as much a permit for your school to hire you as it is for you to work for them.  Notice that the school did not as you said, issue the permit.  The government issued the permit  and gave it not to you but to the school.  The Jacket of mine said "1.  Present this work permit to authorities when requested."  Again as a practical matter the only place they will request it is at the school.  I know that my school was inspected once and had to produce all the permits.  Hence they would need the originals.

If you wish to work at another school, you need to get another permit anyway, but you can take that certified copy in and the officials will be able to look up the pertinent documentation.  Of course, I am assuming that you would get a second permit in the same city.

More important to carry on your person would be the TRC assuming that you will get one along with your WP.