The purpose of protected questions is, according to the message presented on such a question, "to prevent 'thanks!', 'me too!', or spam answers by new users." The more verbose "Protected questions" privilege page agrees on the purpose.

Considering this, shouldn't a user be able to answer protected questions if they have gained at least 10 rep on any stack exchange site? If they've already demonstrated one site that they are familiar enough with the SE Q/A format to not be a drive-by useless answerer, shouldn't that confidence in them carry over to other SE sites?