I have a copy of my great-great grandfather's civil war discharge and pension papers. The copy I have is from the original that he received upon discharge in 1864. I do not have the original and even though I act as the family genealogist, I am not really entitled to the original just because I have an computer with some names in it.
My great-great grandfather's had 6 kids and each of those kids had between 3 and 8 kids and that only puts me at my grandfathers generation, there are about a hundred more people a generation below that. The point is that I am one of hundreds of people who is a direct descendent to him. The record is also in the archives if the copy in the family ever gets lost. I was fortunate enough to talk a great uncle into letting me get one copy of the fragile document many years ago before he died. From that one good copy I made many more copies and scans.
When my great uncle passed away that documents went to his son. I assume they are still with him since everyone should realize the significances of the document. The 1980's cartoon series G.I.Joe always ended each episode with a life lesson and the catch phase "...and knowing is half the battle." This phase also holds true for original documents that you have as much right to as everyone else. Knowing where they are, having and copy for yourself and knowing that they are safe is half the battle, there is no need to fight over getting the original.
Not yours to take
Once things go out 2 or more generations, many more people have a sake to claim on holding the original. While it is imperative to make sure the document or picture are safe and trying to get a copy even if you have to drive the owner to Office Max so they can see you make the copy and give it back to them, you do not have to have the original and you may not be entitled to it. But knowing where the original is, "is half the battle."
If my house ever caught on fire and my fireproof safe where I keep important documents failed, I would know where I could get another copy of the document. Historic documents get lost and destroyed all the time, but if they are in the family and you know who has them, you can sleep well at night with your copy.
If you are concerned about the document, it wouldn't hurt to reach out to the family member about preservation techniques to make sure they are about to pass down the document to their family as well. Originals are wonderful but if 100 people are just as entitled to them as you, then a copy might have to be good enough. Knowing where the originals are and even noting that in your research can give future generations an idea of what family line they went down.
The other side of the coin
There are occasions when people want you as the family genealogist to take the original, since they might not have an interest or storage. This might be true of pictures of distant relative that have not descendants. Great Uncle Sal was a life long bachelor, so when cousin Kelly got a box of his personal items she didn't want to throw them away and wanted you to have the box because you are the keeper of the genealogy.
In genealogy, you can end up with a lot of "stuff", books, pictures, documents and letters. It is important to hold on to the things you get and get copies of the things you don't possess, but if you had everything you would need another whole house just to store it. So be happy with what you have and hope that future generations have the room to store even more generations of family significance.