If the person/company from which you are ordering the part is an Apple Certified Technican or Certified Service Provider, then it is safe to give out. Ask them to show you their certification. However, if it is just some joe-shmow, then there is pretty much no reason he would need your serial number, nor would he have the resources to do anything with it. (You need the Apple Service Software to research serial numbers.) Hope this helps.
Post back w/ questions before giving the information away.
Mike