Do we get fresh water with that lithium extraction?
Desalination plants work with reverse osmosis, which converts seawater to freshwater, with concentrated brine as a by-product. That brine is a better starting point for lithium extraction than seawater, so, yes, they could be co-produced.
But extracting either from seawater does not really make any sense. Some mid-east countries desalinate so they can pursue idiotic schemes to grow wheat in the desert, when they could just buy wheat for far less. California has a few desalination plants, because of dumb policies that vastly inflate the cost of water to urban consumers, while subsiding the delivery of rainwater to farmers growing rice and cotton in the desert.
Likewise, lithium from seawater is not economical, and is unlikely to be so in the foreseeable future. It is better to extract it from salt deposits, or existing brine pools. But the seawater extraction cost is a clear ceiling on the price of lithium, and negates any prediction of a lithium supply crisis.