Whether a person thinks they are being watched or nor is irrelevant. Either they are or they aren't, and if they think they always are being watched, then it is safe to assume that person has integrity because they always act with integrity whether someone is actually watching them or not.
If you always feel like you are being watched then it doesnt matter what you would do if nobody was watching. Youre going to act the same as you always do, and if that is acting with integrity then that won't change.
To me, being a good person extends to doing good while hopping on one leg.
That's ridiculous, right? That's obviously a definition of 'good' I'm creating, and choosing to impose on myself. Which is exactly what you're doing. Yours is just less ridiculous.