Working on Student Visa

[japankyo]

Signed up for a Japanese language school in Kobe for 6 months (which starts soon). I applied for a student visa on my own after I received the "acceptance letter" from the school. I received the visa 2 weeks later. The school is telling me I can't work until after I study with them a certain amount of time. That's their policy NOT immigrations. However, I went to immigration and got a "work permit" (permission to engage in activity other than your status of residence). This involved walking in, filling out a form, and waiting 20 minutes. They gave me the sticker in my passport there.

I'm having a hard time sorting through constant misinformation by other foreigners and the schools. I was told by the school it's impossible to obtain a "work permit" on my own. "You must go through us". This must be a lie or a huge mistake by immigration. Other schools and foreigners were saying it was impossible to get a student visa on my own, etc.

My question is: If I find work, will immigration inform my school? Is there anyway the school can find out that I'm working other than by someone spilling the beans?

Please don't respond if you THINK you know the answer because your friend's friend once heard.... Anyone with actual experience would help a lot. Thanks!!