short answer no....long answer yes...
the user who get's the mail...probably not gonna be able to do a full trace on the mail and get the actual sender's full info....they could in theory get close...like a town...or regional location...but not a name and address....BUT if there were reason to do so...they could get a court order to subpoena the isp records for that mail and get all the information they need about the sender...
your question begs another question....what are you sending that is that offensive/sensative/or plain illegal that warrants you to worry about that?