If you look up 엿 in a dictionary, the first definition to pop up is "a glutinous rice jelly taffy," which seems to have no correlation with the meaning "secretly." As a prefix, "엿-" can also mean "six," as in 엿새, "six days," which, again, seems to have no correlation with the meaning "secretly." However, I also found that 엿 was an old word for 여우 (fox), which is known to be "cunning" or "crafty," and that implies secrecy or furtiveness. Therefore, I am simply wondering if 엿보다 originally meant "to watch like a fox," and if 엿듣다 meant "to listen like a fox"? Again, I do not know; I am just wondering.
UPDATE: Wow! I guess I was right. Immediately after posting the above, I decided to do a Google search of "엿보다." (Yes, I often do things backwards.) Anyway, I found that in a book entitled "우리말 뉘앙스 사전," by 박영수, the 엿 in 엿보다 means "여시," which was an old name for 여우 (fox). It also says that the word 여시 is still used in some areas of Korea. Here is how the book described 엿보다:
'엿보다'는 상대방이 눈치 채지 못하게 몰래 숨어서 가만히 보는 행위를 뜻하는 말이다. '엿보다'의 '엿'은 '여시'를 의미하며 '여시'는 '여우'를 뜻하는 옛말이다. 지금도 일부 지방에서는 여우를 '여시'라고 말한다.If anyone is interested, and I am a little interested, you can buy the ebook version of "우리말 뉘앙스 사전" for $7.39 at THIS LINK. Believe me! I did not post this to try to sell this book. I just feel I need to supply a link since I quoted from the book.