It really depends on what they treated their pool water with. Most pool chemicals are primarily some form or chlorine. There may be some other ingredients but they tend not to be too toxic (otherwise it wouldn't be safe for people to swim in). If the pool has sat all winter without being treated there may not be much if any chlorine left in the water they drained as it does evaporate.

It can kill plants if it's concentrated enough. It can also kill soil microbes and soil critters like worms. However, the good news is that chlorine is acutely toxic but it does not persist. It quickly evaporates or breaks down into other components. Which means it will kill on contact but after a day or two will no longer effect your plants. So basically at this point the damage is done but whatever survives the next few days should be fine. If you replant in a few days the chlorine will not harm the new plants.

The bad news is that some of the chemicals the chlorine breaks down into, like dioxins, are harmful. They will not kill plants but do accumulate in the plants and thus potentially be eaten by you. However, the tiny amounts that may result from this exposure are probably irrelevant in the large picture. If you water your garden from tap water you're adding small amounts of chlorine with each watering and over time that probably adds up more than anything in your neighbors pool.

Bromide is also a common pool chemical, it's quite toxic to algae and fungus, but it won't harm the multi-cell plants in your garden. In fact bromide is a commonly used agricultural fungicide that is injected into the soil to kill fungus, nematodes, and some insects. So maybe you just had a free pesticide application.

The other thing I can think of that could cause problems is that most people add some chemicals to keep the pH of their pool water low (makes chlorine more effective). I'm not sure what effect this might have on the soil or if it could lower soil pH. If there was enough applied I expect it could. Maybe test the pH of the soil in the areas that absorbed most of the water to be on the safe side. If it's low it shouldn't be too hard to add some soil amendments to bring it back up.

Overall, I wouldn't worry too much. I expect that your plants will be fine. If any die, replace them and don't worry about reusing the bed.