Not all need a river to cool but every single nuclear power plant needs a consistent water source 1) to actually generate the energy (steam!), 2) to cool, 3) as emergency safeguard when things go wrong. If you followed the news about Russia's attack on Ukraine and the risks to Zaporizia much of it was about fear of an incident, among others due to water sources or the energy to bring water to the plant being at risk

> 1) to actually generate the energy (steam!)

Secondary loop can be made closed system. This reduces efficiency, but this could be acceptable considering water scarcity. Of course there would be some leaks etc. so water needs to be resupplied, but in much lesser amounts.

> 2) to cool

Not necessarily true, because primary loop doesn't need to have water (HTG, MSR, LMR reactors). HTG reactor actually doesn't need water at all and can power gas turbine directly. LMR, depending on chosen coolant, can do this as well, though it requires heat exchanger.