Usually, people pass it on to their children - or another family member interested in maintaining the family lines. There is no test of worthiness. We record what we know to the best of our ability.
As far as not writing in books - I agree, but this Bible actually has pages in it just for that purpose. Interspersed with the KJV text and hundreds of images are pages asking the owners to list the families' marriages, deaths, births, etc. I guess you could say it is meant to be partly notebook and partly Bible. In the back, someone (probably my 4th great-grandfather since the page is dated August 1887) wrote down a list of addresses.
When I was about 10, someone gave me one, but I have no idea what happened to it. And I haven't seen another one like it, but I know they used to be popular.
I knew of this one's existence, but never thought it would end up in my hands. My aunt left it to her daughter. When her daughter died, it ended up in an estate sale. The woman who bought the estate passed away and her granddaughter thought the book should be back with family and tracked me down on Ancestry.com.
I was astonished to see the family line going all the way from the late 1700's to having listed my mother and myself. It's been maintained for that long. And, as such, I was unable to share many of the pages in it so I don't violate the privacy of the still living.
I have no idea who I will leave it to. My sons aren't really interested in genealogy, and neither are my sister's children. However, she has a daughter who, while she doesn't have children of her own, cares a great deal about the family history. She may not care to maintain it, but at least I believe this Bible will be in good hands.