The boiler side of reactors is not my area of expertise, but I'll try to answer your question. My understanding is the water that is used to make steam for the turbine is a closed loop because the water chemistry and steam quality is very important to safely operating the fast spinning turbines with many blades. For that reason, the water needs to be fresh water.

For desalination, my understanding is there is still enough heat in the steam after it has run through the turbines that it can still boil water, but is not useful for generating further electricity. It is a way to use this waste heat. I believe such systems completely separate the turbine loop water from the desalination water. Essentially, it is seawater sprayed on hot piping to create fresh water steam. The turbine water (downstream of the turbines) flows through the piping in a closed loop to keep it hot.

As for desalination at nuclear plants, there is experience in Kazakhstan and Japan in doing so. I don't know why it hasn't seen more widespread use. I am guessing the need is not that great as lots of reactors are located near large bodies of fresh water by design. The nuclear physics in a reactor can also be complicated by changing temperature, so complicating the heat removal system is probably avoided unless it is a specific requirement.