I'm not going to assume a level of technology here, and instead list a few primitive methods that may lay the foundation for later development in your species.
These methods attempt to localize any grossness to one area of the body, where it could be cleaned with some seaweed or something.
Option one: The Three Seashells
Well, one seashell. A big one.
Providing that your merfolk have all of their relevant orifices located closely together, when mer-nature mer-calls, your merfolk might try to find a large, empty gastropod shell to press up against their body and expel their waste into. When they are done, they quickly cap it with something (another seashell?) and then bury it.
The future: Manufactured waste containers are readily available anywhere in society, along with a nearby receptacle for disposing of them.
Option two: Pissing into the Wind
Let's say your oceans have strong currents. A merperson could park themselves right on the edge of such a current, allowing their waste to be moved downstream with it. It would spread out eventually, but hopefully not before going a significant distance from the merperson. This has the added benefit of the running water cleaning them off.
The future: Basically just a space toilet. We probably should have looked at zero gravity solutions first! Your society builds plumbing networks that allow high-strength vortexes to be created at the endpoints. Then, it's easy to build toilets that suck away waste continuously while active. The pipe that refills the toilets could have a receptacle near the endpoint for disinfectant containers that clean off the merperson when they press a button. This water would then also be slurped up by the system. This is probably the cleanest solution.
I made this a numbered list because I anticipated having a third idea, but then I didn't. If I come up with a third unique solution, I'll add it.