Yes they do. The white-hat always announces this in the NFL. If you listen closely during an extra-point or field goal you will hear WH announce it. They don't put the audio on the tv but only announce it in the stadium. This way the defense and officials are aware of who is declared legal. It is a 5-yard penalty to have an illegal number in a legal position and not declare it prior to the play.
I got this off of this website
http://ww2.nfl.com/fans/rules/
If a player changes his eligibility, the Referee must alert the defensive captain after player has reported to him.
I believe these aren't the "real" rules from the rules book but instead a summary of them. WH informs the defensive captain by announcing it over PA.