The short answer is yes. Children typically begin to experience embarrassment in these types of situations around age three. On a related note, I have heard from several parents that I know that children typically begin to blush and/or cry when they have an accident in their pants around age three to four.
I also read recently (but I don’t remember where) that if a child reaches the age of three-and-a-half to four without experiencing embarrassment in situations like this that it is one indication that he or she may have autism or some other developmental disability.
I have actually seen a couple examples of this in real life. A friend of a friend of mine has a five-year-old grandson who often wets in the front yard of his grandmother’s house with no apparent sign of embarrassment. I have been told that he is rather severely on the autism spectrum. His older brother (around a year older) is also on the spectrum and he did this until last year without any sign of embarrassment. At least now when he has an accident he is usually embarrassed by it (Now he usually blushes and cries when he wets). They were both diagnosed around age three and thanks to some speech therapy they are at least verbal now (on average to good days). Also, physical therapy has helped them to walk better. They don’t fall down nearly as much as they used to and they can climb stairs now. So, in short, screening for conditions like this is worthwhile whether it helps with toilet training or not.
In short, if a three-year-old is embarrassed when he or she needs changing it is normal and a good sign that he or she is coming along well (developmentally speaking).