Ours is done in the way I prefer providing that direct overboard
discharge of grey water is allowed in the region sailed in.
Shower and bathroom basin waste goes to a small ss sump tank (which needs a vent, we take that back to the shower compartment in case of an overflow, which has never happened) that has a SureBail float switch in it. The pump out is an external multiport pump for grey water (SureFlo in our case, but ours maybe now a discontinued model I think) with a strainer before it. Pump goes directly to the grey water discharge seacock manifolded with the kitchen basin wastes as well (so in that case we have to be careful that the seacock is always open when shower/bathroom basin used else the grey water from the sump gets pumped into the kitchen sinks
The ss sump is sized in plan area so that with the shower and basin taps running the lift
in the float switch between off and on cycles the pump fast enough so that the sump does not overflow - resulted in a surprisingly small tank for us (less than a foot square). The level in the sump therefore fluctuates between the off and on levels of the SureBail switch. The sump tank has a clear handhole in the top (similar to those used in sailing dinghy
The discharge from the pump does not go through a vented loop even though the shower pan is well below the waterline - the valves in the multiport pump prevent backflow and the builder
believed this was enough, but although I agreed with him I had him put a bronze non return valve in the line as well, just in case. If there were to be backflow it goes back into sump and the pump cycles to pump it out again - we leave the pump alive all the time we are on board. The reason we did this was due to the difficulty of finding a place to put a loop where it would vent into a wet area should its vacuum relief valve dribble - in a bigger boat or a small one with wet bilges or lockers not minded if get damp it may not be an issue.
The only issue is the strainer before the pump fouls very rapidly especially with woman on board due to hair even though we used a coarser strainer than the pump manufacturer recommended. In future I would put a removable ss gauze in the pan drain hole to act as a coarse first cut at the solids (another alternative I've seen is in the same place or inside the ss sump tank itself a holder for a ss pot scourer - the type that looks like turnings off a lathe - which acts as a strainer and is chucked away when fouled).
The sump seems to self clean. If you have a water trap in the shower pan drain (U bend type thing) rather than a thin rubber joker valve to stop smells coming back then the vent in the sump has to be large enough to take the basin drainage displacement
of air without the water in the shower trap blowing back into the shower tray. We had a water trap but removed it as we could only fit a small diameter vent and we had that vented into the bathroom anyway - smell hasn't proved a problem except when the boat has been unused for a few days, that because the small sump is constantly flushed and does not act as storage
of grey water. In positioning the sump one needs to keep in mind boat motion and heel and its effects on inflow, outflow and float switch operation - from memory, without reinventing it, I think this meant inlet, outlet and float switch are best on the fore and aft centreline of the sump tank.
In 12 years we have not had a failure of anything except I had to remake the soldered and lined heat shrinked 12v connections to the float switch earlish on due to the wet atmosphere in the sump but I remade them with lined shrinkwrap with a good dollop of hot melt glue added inside it for good measure and no further problem. But we do not do charters and few non boaty types ever on board so abuse is not a problem for us - use includes having lived aboard for an around 10 month period, and on board usually 2 days almost every week and cruising near 2 months per year.
Although I have never used one, I would personally be wary of a submersible pump inside a grey water tank - can't think of a really good reason why I feel that way except is easier and cleaner to maintain if outside and will not get covered in soap and body fat slime
. Same with the strainer - in line outside means pretty clean to handle.