Personally, I do find embarrassment an essential part of spanking. The three things that cause me the most embarrassment (in no particular order) are
1. Other people either present or within hearing distance of the punishment. This, to me, is the ultimate embarrassment. Especially if they comment on my squirming and sobbing.
2. Verbal Chastisement during the spanking. Several spankers have been so good at this part that my tears were more from their words than my physical pain.
3. Corner time afterwards. Hands on head, red bottom available to anyone who may happen by. Very embarrassing.