@Anne: As a private piano teacher, I dipped into my own resources to buy music, keep my piano tuned and in excellent condition, pay music teacher association membership dues, etc. The amount I charged for lessons didn't cover all of my expenses, and Uncle Sam took a big chunk as well. If you make over $400/year, which isn't difficult to do, you have to pay a self employment tax on top of the regular income taxes you already pay. Because of that and because of the tax bracket my DH and I were in, at least half my take-home pay went straight to the government. I went back to working in an office where I have a steady salary and fringe benefits.
I personally would buy a gift for a piano teacher. I always bought gifts for my piano teacher, who instructed me year 'round for 11 years. That said, I don't think it's necessary, and I don't think piano teachers expect you to do it. All of the ones I know are like me--they just want respect and appreciation.
Hope this helps.