Clearly, every billboard location is different and subject to different state and local ordinances. But the net effect is the same in 99%+ of all cases. Here's why. If the sign is legal, non-conforming, it cannot be moved or altered, only removed. So if the landowner and the billboard company cannot agree to a ground rent, then the sign is torn down and the income is lost for both parties. If the location is legal, conforming, then all the sign company has to do (assuming that the spacing, zoning and ordinance allows it) is to file a demolition permit and a simultaneous new application on the neighboring property, which no other sign company or individual can get in front of as the sequence is that the first permit filed after the demolition application is in the first position. Essentially, the sign company holds all the cards at all times.
In almost all states, the landowner signs the permit application, but that does not give the landowner any ownership of the permit. Substantial case law supports this. They sign the appliction to prevent fraud by the sign company filing for permits without the landowner's approval. Up through the 1980's, most states did not have this requirement, and the billboard companies would pull permits on every legal location (without the landowner knowing it) and then use the permit to blackmail the landowner into signing their lease or moving the permit to the neighbor's property -- which is why they changed the system. In Canada, the case law puts the permit in the hands of the landowner, which is why so much of the outdoor in Canada is wooden, since the sign company does not want to put any significant investment into something they do not control.
The reason that the sign company did not buy the land next to your sign is because almost all sign companies have no interest in owning the land where the sign is located, but prefer to lease it. I did not know this myself until several years into my billboard career, when I was mentored by the head of real estate for Foster & Kleiser in Dallas (now Clear Channel). He explained to me that owning the land only hurt the sign company's position as it forced them to deal with new issues such as property maintenance and insurance, and also reduced their ability to re-negotiate and even abandon leases that were no longer profitable. In addition, it required large amounts of capital that (based on rates of return) could not be justified when compared to just paying rent (for example, a $6,000 per year groundrent could not justify buying a parcel for $400,000). If this was not the case, then Foster & Kleiser could have bought nearly every parcel their signs were on in the 1950s through 1980s -- but instead they only owned a tiny fraction of them (maybe 1% or less). Additionally, most outdoor companies today have a skeletal staff in the real estate department, and would not have the time to research property going to auction, or the manpower to go to such auctions. The modern real estate department only handles lease renewals, permit problems, and ground rent renegotiations.
These are just the facts as I know them. But again, every location is a separate case, and you should do decent due diligence to see what your rights are.
The key item here is that all landowners and billboard companies need to work together for a win/win solution. Even without free-market competition, you can still have a situation in which both parties respect each other and come to a fair deal for all.