If you go deep enough, sea water will permeate the whole land mass. The depth of the boundary between salt water and fresh water depends on many factors such as the hydrogeology (rock types, permeability, porosities, hydraulic gradients, structural geology (including faults and fractures), etc.), precipitation rates over the island, and the relative densities of the sea water and fresh water. For an aquifer characterised by a homogeneous, isotropic porous medium, the 3-dimensional shape of the aquifer can be visualised as a lens of fresh water that sits above the salt water. The transition zone between the fresh water and salt water is usually characterised as a mixing zone, where salinity increases with depth. For more complex groundwater systems, the relationship between fresh water and salt water can vary greatly. For example, on volcanic islands, it is not uncommon to have confined aquifers that are isolated from the sea water by dikes and layers of volcanic rock.
The fresh water originates as precipitation on the island, which recharges the aquifer(s). Groundwater flow paths largely depends on hydraulic gradients and the hydrogeologic characteristics of the aquifer. It is certainly possible for groundwater to discharge into the sea via preferential flow paths such as faults and fractures.
I hope this helps.