If you have standard gravity-flush toilets. They aren’t directly affected when the power goes out, provided water is still flowing and your waste system doesn’t rely on electricity.
If you get your water from a well, because you also have holding tanks. While water pressure may be reduced when the well pump and pressure pump lose power.
If you have a septic system that uses an effluent pump to transfer waste from the tank to a drainfield, or secondary treatment system located at a higher elevation. The pump won’t work when the power goes out.
If you have upflush toilets that plug into a wall outlet and need electricity to operate macerating blades, and a pump that transfers the waste to the sewer.
IIf you live in an apartment building. You might lose water during a power outage because the pump that circulates water throughout the building stops working.
Once the power goes out you will have at least one flush available so try to save it for when you really need it. As the saying goes: If it’s yellow let it mellow, if it’s brown flush it down.