Depends on if they use evaporative cooling or not. But the water returned is warmer which can harm the ecosystem quite a bit, both for power plants and data centers.
The heat issue is also widely misunderstood. The state of California gets on the order of a PWh of heat every day from the sun. A million GWh. Even a large nuclear plant doesn't make a dent in that. The issue is localized to the immediate area at the outlet, because the water temperature directly at the outlet is going to be higher than it is at the inlet, before it diffuses back into the background. Naturally the way to mitigate this is to pull more water through the cooling system, all of which you're still returning, but then you raise the temperature of twice as much water by half as much temperature, which is what you want since the temperature delta is the concern. Of course, doing this will cause people to accuse you of using more water, even though you're neither consuming the water nor modifying the amount of heat being transferred to the environment.
In addition to that, water has extremely different levels of scarcity depending on what region you're in. Parts of California and Texas have arid climates and extremely limited economical means to source additional fresh water. Areas like the Midwest get dramatically more rainfall and as a result fresh water is not meaningfully scarce there.
No, not necessary to the same one. Every body of water has a catchment area, and through the bottom parts of some lakes/rivers water sips into the earth, filling underground water levels.
Now, let's say you life on a coast with a regular winds in the ocean direction.
You basin fills with water evaporated by you, your neighbors catchments and – sometimes – ocean (wind doesn't always blow in one direction). But the large percentage of the water you evaporate goes into the ocean. Let's say 90 percent falls again on your catchment area, 10 percent falls in ocean but is regained through neighbors basins evaporation.
Now, if you increase your evaporation, you could easily drain a lot of water from your land.
Hypothetically you could recondense it somewhere else, i.e. get a large land area and then capture the rainfall. Probably the big win there would be to take the water which is currently routed into the ocean from storm drains and recover it instead of doing that with it.
At which point we're back to the quadrillion watt hours of sunlight you get a day, which evaporates such a large amount of water that you have to be doing something pretty dramatic to move the needle. Which thermal power plants and data centers don't, but agriculture? It's pumping trillions of gallons of water out of the ground and into fields in that sun.
Most of the heat doesn't go into evaporation. If it did, we wouldn't have higher temperature during the day, just more moist.
Truth is, I personally think farming evaporation is leagues more ahead of datacenters. Datacenters are more of a nuances locally, where depleting resources already don't give enough water to keep households water pressure up. Add to that a new water-hungry datacenter, and pressure in pipes drops even more.
They aren't bad on their own, but in a context where they want to share a depleting communal resource.
Edit: tl;Dr farming is worse but historically there. There is a lot of discourse around that, also, but it's a separate issue.
This is about as useful as saying "the water stays on Earth". The ecological damage entirely depends on the nature of the water source.
The most poignant situation is non-rechargeable aquifers. When water is extracted, the mineral structures collapse and the storage capacity is forever destroyed.
And as the other reply stated, evaporation (as from evaporative cooling) exports water out of a catchment basin so that it can't be used downstream, just as if you pumped it away.
That is actually an important distinction. There are many things for which this is not the case. If you fail to recover the helium extracted with natural gas deposits then it doesn't stay on Earth. If birds eat your crops, you can't undo it. If you burn coal then it turns into CO2 and heat and it's thermodynamically impossible to reverse the process without inputting more useful energy than can be recovered from burning it to begin with (which is why carbon capture is a sham and we instead need to stop burning fossil fuels).
Whereas you can evaporate and recondense the same water an unlimited number of times and whether it's a problem depends not so much on what you're doing as where you got the water to begin with.