As to the reasonable concern:

"When I have been badgered into describing experiences I've had that can't (yet) be measured in a scientific sense, I've been asked to prove the reality of the experience to the skeptic. "

It's all in HOW you report these things. If you simply say "I had this strange experience and I wonder about it and what it means..", even *I* won't harass you! But when you suggest that it must BE true and that NO other explanation is possible...and often, that others who have NOT had such experiences should accept it and act in some way as a result, you do get resistance.

I would never assert that someone did not 'seem' to have the experience they report. But I am not willing to agree that you are the reinacarnation of Ghenghis Khan or that you were abducted by aliens or that we should evacuate Schenectady because you 'saw' it burning in your dreams. Yet...many believe everything Nostradamus and others said...

If it ain't the sort of thing that can be proved, be aware of how you present it.