A sink doesn't need any power source or batteries to work properly and can function fully during a power outage.
The answer to this question is yes, it does; but indirectly. Considering the process of how water flows from the faucet through the pipes into the drain and out of your home, we know that certain important appliances do use electricity such as electrical appliances like a waste removal system. Some other factors that determine the usage of electricity are the plumbing system of your house or affiliated sites, the usage of an electric pump or sump pump, the sewage system and so on.
If no power source is present in the house or when the power goes off (during a power outage), fresh water should still flow freely through sinks and drains through its water pumping system through gravity. This is because faucets have an aerator attached to the end of a pipe(s) connected to the main supply line of city water or the site water. The aerator speeds up the flow of running water but also acts as a valve that can close off or open depending on the main water shutoff. The main supply line connects to the stop valve that may need some turning now and then to ensure a free flow of water out of the faucet. It also connects, through another pipe, with an electrical discharge device/switch that when flipped on, sends power to the entire drainage system. This ensures that water flows in both directions out of your home into the pump assisted sewer systems or septic tanks in case of a power outage. Contrary to the gas water heaters (which can heat up the water in few minutes or a little while during power outages when the power goes off), the hot electric water heater needs power. So, unlike a heavy-duty trash bag, you will need electricity back up to use water heaters. There are several risks associated with electric system usage in plumbing systems likes property damage, improper functioning during a power outage, overloads, breakdown of the sump pump, etc. Sometimes fixing these issues requires the usage of specialised items like plumbers lighting candles and therefore you should let a licensed electrician take care of such plumbing risks.
Showers have much simpler plumbing than sinks when it comes to electricity. The water pressure that comes from a shower head can be generated by a pump. Since there is still some water flow in the pipes, taps and faucets activate an electrical switch/pump (which every home plumbing should have) that sends electrical power from your main plumbing line to activate & power most components needed for showering (including hot water for a warm shower). It also lets the water stay warm inside its tank. Whether you have a tank-style water heater (with large containers) or tankless water heaters, it is important to note that both run on electricity just like your air conditioning system. The level of electricity consumption in a tank-style water heater also depends on the amount of hot water, and temperature, left in the hot water tank.
Most toilets do not use electricity and can function during a power outage. Pump assisted toilets work without electricity because it uses its own built-in pump in the tank that sends water down into the toilet bowl. It would directly flush the toilet water out of your home through a septic tank or sewer just like what sump pumps do. It means even when there is no electricity or a power outage, you can manually flush as much water as you need with the integrating plumbing mechanism. The only time electric power is needed for a toilet is when anti-flooding mechanisms, bidets or air blowers are used for issues like the clogged pipe. As far as the concept of a good old toilet flush is concerned, it does not use electricity and its functioning depends on how much water is left in the toilet flush tank.
In most cases, bidets are connected to the same water supply lines as other fixtures in your home. This means that when you turn on a faucet and wait for the hot/cold water to flow out of it, it also flows into the bidet(s) at the same time. If your bidet is not powered by electricity but by some sort of pressure mechanism (such as pressurised air), this may cause water damage if no power source is present in your home. However, these devices are normally much more expensive than installing an electric-powered one so they do not face any major issues with being unable to operate properly without electrical power. Also, since they have their own built-in pumps, toilets usually don't run into problems with bidets either.