The desalination part would be in a continuous state of sinking, pulled down by the weights. However, it would never actually sink as the fresh water is pumped away. The pump could be powered by a solar cell and wind turbine, coupled with a battery, or alternatively by a power supply from shore.
The fresh water could be pumped directly to shore, or to a reservoir. The reservior should float as fresh water is less dense than salt water. It may be possible to achieve a balance such that the reservoir provides buoyancy for the desalination element in the event of pump/power failure.
The pipe connecting the desalination element should be flexible to allow relative movement of the two.
Plants could be anchored or permanently affixed to the sea bed. Plants anchored in shipping lanes could be used to provide emergency fresh water to passing ships.