What information do I need to size a sawn or hewn beam?
It depends on what kind of wood you're using. Different woods have different strength characteristics. You may want to swing by the local library and grab a copy of a "Timberframing: Design, Construction, Finishing" by Ted Benson, which has tables of wood characteristics. 

Note: If you live in a town where a permit is required to do the work you're planning, you're going to have to get sign-off by a structural engineer - in which case, you might as well ask the engineer what dimensions you need for the wood you're planning to use.