French press coffee grounds should be disposed of either in the trash or a compost pile. Never wash coffee grounds down the sink, if possible.
Coffee beans contain essential oils that remain in the grounds after the coffee is brewed. These oils could clog a sink over time, costing hundreds of dollars to fix. This is why they should never go down the drain. Small amounts are okay as long as soap is used to break up the oils.
Unless the French press coffee maker has a rubber bottom that comes off (some do), the only way to get the grounds out is to scrape them with a wooden or plastic spoon. The little bit that is left can be washed out with soap and water.