Have you heard of the saying: "Extraordinary claims require extraordinary evidence"?
It isn't true that the weight of evidence lies equally with both sides when making a claim. It is up to the claimant to provide the evidence which is then refuted or not refuted based on the quality of the evidence provided by the other side of the debate. and so on...
BTW, logic also demands that one not insist on physical evidence for non-physical reality.
If the non-physical 'reality' is able to affect the physical reality, then it isn't unreasonable to ask for physical evidence of said effect.