Christopher Nolan has complete involvement and final say in his trailers, but no, most directors don't get final say. Many get to choose from a few iterations, and make suggestions here and there, but unless you're Nolan/Spielberg/Fincher/Cameron/etc., your input is minimal.
And most studios outsource their trailers. They'll give two or three companies all the footage they want to use, or let them have/see a working version of the movie so far, and then choose their favorite cut of the trailer amongst the companies. Sometimes they'll ask that companies stick to a certain theme/idea, and other times they'll let the companies get as creative as they can. Either way, for some trailers, it's almost like a competition among companies, and for others, studios have one company/editor they really like to use for certain projects.
[This message has been edited by TCTTS (edited 5/6/2014 12:25p).]