The law is full of twists regarding this issue. Reading the FL Supreme Court case of Tona-Rama, IMO, it sounds as though the property owners do have rights to their land and the public does not. The Court states, in Tona-Rama (other cases may vary), that the public has the use of the beach, but not rights to the property(beach). The public cannot set up a vending station or build a seawall, etc. because they have no ownership of the private beach. In my non-legal opinion, I think it could be said that the state, county, nor TDC, have the "right" to nourish the beach, but the public may still be able to use the beach (dry) as they have for many years.
Now, if the property owners refuse to have their property nourished, it sounds as though they maintain the rights to the property. With those rights, also comes liability. Who knows what is likely to happen and who is likely to win cases if a hurricane funnels water and damage to the property owners who forgoe nourishment. It is likely that they will file suit against the State/County for nourishment to the neighboring properties which caused the funnel of damage to their non-nourished beach. If neighboring properties located behind the non-nourished properties become damaged due to the funnel-effect (my word), they will likely have a good case against the property owners who didn't accept the nourishment. They would likely go after the State/County as well. Nourishment and seawalls (similar cases will be likely) is a trainwreck which will happen with time.