There’s no reason parents can’t start to sign with newborn babies, and in fact, some do. While a baby won’t respond until they are at least 4-6 months old, they will still receive some moderate benefit. Some babies respond in a receptive way, that is, they understand what is being said, much earlier than most might think possible. This is why many experts promote talking with and around your baby, no matter how young they are. Talking around your baby habituates them to the sounds that typify language. Signing around a newborn baby serves the same function.
Babies have been noted to sign back, called expressive language, as early as 4 months in cases when they have been signed to from birth. Keep in mind that most babies will sign back at around 10 months and some even later and usually within just a few short weeks so long as signs are introduced following a regular program.
Therefore when you decide to start signing has more to do with your goals as a parent. Do you want your baby to be signing back to you quickly, or are you willing to sign by yourself for a while first? Most parents find that having some feedback is important to stay motivated and a parent simply wont get anything expressively back from their baby until they are at least 6 months old, with the usual more like 10-12 months. What you decide to do is a purely personal choice.
My personal recommendation for the first 9-or-so-months of your baby’s life is to familiarize yourself with baby sign language and its rules, learn how to do some signs and practice them in anticipation of making a big push when your baby seems ready to fire on all cylinders.