There is no dissolving them again. Ions remain dissolved all the time.
It is like taking
- $\pu{11 m3}$ of sea water with salinity e.g. $\ce{30 g/L}$,
producing
- $\pu{1 m3}$ of (about) pure water (with residual salt)
- $\pu{10 m3}$ of sea water with salinity $\pu{33 g/L}$
by active redistribution of water (like in reverse osmosis) or ions(like in electrodialysis) between 2 separated volumes. All numbers are just illustrative.
It is technologically/economically advantageous in desalination to use big excess of input sea water to keep the salinity difference between the input water and the waste water small. With big difference, the process would be less efficient and more demanding.
This is kind of analogical to efficiency of heat pumps, used for building heating. Maximal efficiency is reached at the minimal difference between interior and exterior temperature.
There would be just minor difference between processing of the waste water from desalination and raw sea water, if we wanted to get minerals from the waste water. Additionally, there would be large water volumes, questioning capacity to process it and demand for its output.
For curiosity, here is the Wikipedia table of energy consuption of chosen methods:
See Desalination Technologies for the extensive list.