I imagine brooms would work much like wands do. They are magical objects in and of themselves, but they require a witch or wizard who can use their own magic as a channel in order to use the objects to their full potential. I believe J.K. has said in interviews, were a Muggle to attempt to use a wand, the results could either be miniscule or detrimental (think like in the wand scene in the first movie), but they wouldn't be able to use the object to the same extent that a wizard could.
I would think that if a Muggle were to try and use a broom, either it wouldn't work or it would take off flying without the passager...or with them hanging by three fingers and being dragged through the canopy of a forest or the rooftops of a city. Which do you think would be more painful?