Waste Management

Waste Management on a naval ship is called CHT (Collection, Holding, and Transfer). There are three types of disposables on a ship: waste, soil, and solids. Solids are all the things you would throw in your trash such as food waste, paper, plastic, metals, and all that stuff. That is sorted and put in a special room. The paper stuff gets compressed down and all the solid stuff waits until the next port where it gets passed down a chain of crewmembers to the proper dumpster on the pier. Waste is from things like showers and sinks. That is usually recycled back into the ocean unless a country specifically asks that ships not do that near their land, in which case it is held until it can get pumped into a sewer system in port or the ship is clear of their area. Soil is what comes from toilets when flushed and that goes into tanks. Occasionally those tanks are slowly emptied into the ocean water to become part of that ecosystem in a small percentage but often the soil is kept in the tanks until it can be released into sewers at ports.
It seems wasteful to just dump trash in space as the ship goes by. Not only is it difficult to determine where the trash will land at high speeds (if you decide to do it this way I’d recommend waiting until the speed is slowed to discharge) but there is no way for space to recycle the trash. If solids accidently fell over the edge of the ship in water (as happens in many private small ships) the water will eventually reclaim the materials and reuse them. (True, that often takes years or decades.) In space there is nothing to corrode materials so they will merely float in place for centuries and beyond (by modern understanding on space). As to waste, the ocean merely adds it to the salt water there, what little dirt or sweat there is gets strongly diluted in the ocean to be harmless to the environment. If soil gets in the ocean, it also gets diluted instantly to harmlessness, unless it is near land in third world countries that have enough of that type of soil in the water to not allow foreigners to swim in their beaches. In space you will likely need some sort or recycling system, which is why I highly recommend a replicator system for long voyages. As for soil and waste, a water purification system would work well. All you need to decide about your fictional ship is if the dirty stuff separated from the water is reused somehow, burned to ash to store easier, or merely compressed and disposed of by special waste ships that transport it to waste planets or shuttled down to planets to decompose, possibly changing the ecosystem in the process.
Another type of waste you will need to keep in mind concerns the air. Humans need oxygen to breathe safely but we breathe out carbon dioxide. Even as big as any ship is, it will need to create more oxygen to keep everyone healthy. One way to do that is through a purifying system using water, but it could also be done electronically on ships if the right technology exists. How will you deal with all the wastes on your ship?