Are there terrestrial plants that thrive on sea water?
The University of California, LA mentions some genera and species of salt-loving plants (halophytes):

The genus Atriplex (Family Chenopodiaceae), saltbush, is found worldwide along saline shorelines. On the surfaces of the leaf are vesiculated trichomes (hairs). Each trichome has a stalk and a balloon-like tip, the bladder cell. The leaves use the bladder cells to excrete salt. 


Coastal saltbush. Source: Enviro Active Australia.

Excreting salt glands occur in numerous unrelated plant groups, including grasses that live in alkali sink and salt pans and in warm temperate to tropical salt marshes (Sporoboleae, Chlorideae, and Aeluroideae (Family Poaceae)). These include such well-known tidal saltmarsh plants as the cordgrasses (Spartina alterniflora, S. patens), alkali grass (Puccinellia phryganodes), saltgrass (Distichlis spicata), and shoregrass (Monanthochloe littoralis). Short, two-celled trichomes have evolved as collecting chambers for salts, from which salts are excreted through tiny pores.


Cordgrass in the Chesapeake Bay. Source: Chesapeake Bay Program

Frankenia (Family Frankeniaceae) is a genus found both in coastal salt marsh and inland saline habitats. Saltcedar (Tamarix, Family Tamaricaceae) grows around the world in arid and semiarid regions. All of these taxa have salt glands, i.e., trichomes, in which salts are accumulated and excreted.

Several common mangroves secrete salts from leaves. So much crystalline salt can occur on the leaf of black mangrove, Avicennia germinans, that it appears whitish. Aegialitis annulata is a salt-excreting mangrove of Family Plumbaginaceae, a group in which other species are salt excreters in salt marsh and oceanside (Limonium and Armeria) or have chalk glands and live in dry, alkaline habitats.


Black Mangrove. Source: PlantDex.

Halophytic succulence is demonstrated in many genera of plants that inhabit saline environments, such as the mud flats of coastal salt marsh, but also is commonly observed on the leading edge of vegetation on coastal beaches, i.e., in strand vegetation. Common examples of succulence can be found in Family Chenopodiaceae: Allenrolfea, Arthrocnemum, Halimione, Nitrophila, Salicornia
Suaeda.

Succulence (fleshiness) also occurs in other famous and widespread species, such as Batis maritima, Sesuvium portulacastrum, Trianthema portulacastrum, and many Old World species of Zygophyllum. True succulents in the ice plant family, Aizoaceae, are also commonly found in saline habitats: Aizoon and Mesembryanthemum. By depositing salts in vacuoles, the toxicity is partitioned from the cytoplasm and organelles of the cells. As in Atriplex, salts are removed from the plant when the leaf of stem segment is shed.


Halophytic succulent Trianthema portulacastrum. Source: Altervista.

Among the halophytes, the seagrasses, which are submerged plants of shallow marine meadow, have salt tolerance, because the osmolality of cytoplasm is adjusted to match the osmolality of the seawater, so that desiccation of the plant does not occur. This contrasts the strategies of excluders and succulents, which are primarily salt avoiding mechanisms, attempts to reduce salt concentrations within the cytoplasm of photosynthetic cells.

As to why some shores may be devoid of plants, it may have to do with extreme temperatures, lack of nutrients, phytotoxic material etc.