I had a very embarrassing ride with two friends last night. We were jumping in the front arena and a bunch of kids gathered at the rail to watch. Immediately, Gracie decided to refuse to jump altogether, so did Scottie. The third horse, Funk, just forgot how.
Did they do this to "embarrass" us? Likely not, but there is definite anecdotal evidence that horses do know when they're being watched. Trained show horses know that they will get social approval from their handler when they do well, and in classes where appearance is important they will learn to turn on a bit of extra sparkle. And some horses do seem to get stage fright and perform distinctly worse when being watched. (Mares, of course, are notorious for waiting to foal until nobody is looking).