It depends entirely on where the fossils were dug up. In the US fossils on private land belong to whomever owns the mineral rights to the land, that landowner can do with the fossils as they will. On public land the fossils belong to the government and to get a permit to dig them a paleontologist must have a place to store them, this is part of the permitting process (digging on public land without a permit is a felony). Most of the time this place is something called a repository, a public institution that has facilities to store and care for fossils, usually a museum or university and has contingencies so that the fossils will not be lost if the facility were to fail or close. Now this is for vertebrate fossils becasue of the nature of invertebrate fossils (chalk for instance is made entirely of fossils) the laws can be a bit more lax.
Private collectors can only collect on private land, which usually involves making an agreement with the landowner. Keeping collection records is always a good idea if for no other reason than to show where a fossil was found.
I have worked with both types of collectors private and public, good private fossil hunters will differentiate rare fossils and common ones and will donate rare ones to museums while selling common ones. Bad one will sell everything, which often does lead to fossils disappearing or in some cases being outright destroyed by those who disapprove of evolution and the fossil evidence for it. For this reason most journals will not accept publication made using private collections, becasue it makes repeatability impossible.
Now this is all in the US, other countries have different laws some more strict some less so.