You definitely risk liability. You crash an airplane with a student on board you should expect that you will be held responsible. And
being incorporated would not help.
But you
might be a little paranoid if you worry about it too much and it keeps you from instructing. Understand the risk, evaluate your personal likely exposure, determine your insurance needs, figure out your personal risk tolerance (each of ours is different), and act accordingly. "Acting accordingly" includes not only insurance, but engaging in best practices to minimize your exposure.
For example: I gave a talk on instructors and insurance a few years ago. There were 20-30 instructors in the room, some of which had been teaching for years, most of whom I knew personally. I asked, "How many of you personally know someone who had a liability claim made against them." No hands went up.
I'm a bit of a veteran when it comes to CFI liability discussions online. My personal, anecdotal, and completely unscientific analysis (which means I rely on it but you can't) is that our liability as instructors are, in order of highest to lowest):
1. When we are in the aircraft. Pretty obvious, but this seems to be the one that most CFIs don't seem to worry about.
2. When a student pilot is soloing.
[Big Gap. Cavern, actually. But these are the ones that CFIs tend to be most concerned about]
3. The day after a BFR or IPC.
4. The day after the pilot passes a checkride based on our recommendation.
5. Sometime later.
I've been running an on-line challenge for about 10 years now: Tell me about a real situation when a CFI was successfully sued because of something that happened when the CFI was not in the airplane. No urban legends allowed. No, "Well, I heard about this guy who was the third cousin of a nephew of my ex sister in law who..."
I've managed to collect a total of two solid references.
First is a case from 1958. A flight school was successfully sued when a student pilot took off with the control lock on.
The other: In response to my "challenge" someone mentioned (and I was able to verify) an NTSB determination of "impropy instruction" as a probable cause of an accident in which there was a fatality. I count it because it's highly likely that the CFI was sued or that his insurance settled (unless he was totally uncollectible).
So that's about it. Two real examples over the last 36 years.
I'm still collecting. Anyone here have one?
That does not mean it doesn't exist - just that the risk is probably smaller than most of us think is is.