Facts are for the courts to determine.
Just to flesh that out a bit, because that statement conflates several abstract notions.
In the prosecution of any case, criminal or civil, the burden of production of evidence doesn't lie with the court. The court sits in judgment and oversees the process of the parties presentation of evidence. The court does not present evidence, or make argument.
The "determination" of a fact also involves weighing competing narratives or evidence, trying to get to the truth. Generally speaking, if there is no dispute as to the facts, and the facts / accusation / claim satisfy all the elements of the crime (or tort, in civil court), there is no trial. The point of a trial is to air and decide fact disputes - and a fact dispute involves both sides making specific contentions.
"You conspired", "I did not" is not a contest of specific contentions, any more than "You broke the law," "I did not" is.