contact info really isn't near as important as knowing where a CM would bring a lost child as well s teaching your children what to do if you get separated.
With young children go to a mom with children - bigger small children to a CM. Once a child is a bit older you can say if we get separated we meet at this location (my family always met at the carousel).
At MK lost children go to city hall (last time I checked), at each park there's a different location. As a parent missing a child quickly look in the immediate area, if possible have one person stay there, and the other go to a CM to report the child missing. Disney works fast and most of the time your child will be found before anyone looks at the identifying tag let alone calls you.
With small non-verbal children it could be useful - but you give your children a lot more power if you let them know how to take care of themselves vs. tagging them.