How you proceed when discovering a prospective fossil often depends on where it was found. As far as identifying the specimen goes, it is best to submit as much information to a museum or official paleontology group as you can (including its location) so it can be compared to any other finds in the area. In terms of ownership of the fossil, where you found it is again crucial. If the land you discovered it on belongs to you, then the fossil is yours to do with as you wish (via Smithsonian Magazine). However, if it is on someone else’s property or federal land, it will at the very least be an uphill battle. 

In the case of Sue the Tyrannosaurus (the largest and most complete T-Rex specimen ever found), multiple claimants emerged including the Black Hills Institute (who found her), the rancher whose land she was on, the Sioux tribe, and the federal government. In other cases such as the Judith River Dinosaur Institute, efforts to conceal the true excavation site of fossils they had found led to the group’s head being imprisoned for theft. That being said it is not impossible to win fossil ownership cases. This was learned by the Murray family, who after 14 years won ownership of the Dueling Dinos fossil from the previous owners of their land (via ScienceMag).