I've had decades of experience with boats and RVs, including driving commercial boats. It really comes down to a couple options:
Toilet paper - there is the "rapidly dissolving" kind and the "industrial strength" kind. Walmart (in their RV section), camping, and marine stores sell the "rapidly dissolving" type... which is generally about twice as expensive as what you'd buy at your local grocery store. Besides the Costco brand (not everyone is near a Costco), Angel Soft toilet paper breaks down as fast as the more expensive RV/marine toilet paper. If you aren't sure if your toilet paper breaks down, put a few sheets in a jar of water, shake the jar... in a couple hours, if you can still see whole sheets, it isn't "rapidly dissolving."
Another issue: can you trust the people using the toilet? Driving commercial boats, probably not. Oh, you can tell them to "go easy" with the amount of toilet paper they use, but you will still have the "wadders" who use a couple fists full with each wipe... guaranteed to plug up the hardiest marine head. For that situation, I go with: if you didn't eat it or drink it, it doesn't go in the toilet (and that includes feminine hygiene products)... if you are worried that someone else is going to see your toilet paper, pull off a couple sheets and make a nice little cover for it IN THE GARBAGE CAN. I let folks know that they can use all the toilet paper they want, they just CAN'T flush it down the toilet.
On your own boat, with your own family, if you can trust them to be conservative with their (rapidly dissolving) toilet paper use, flush away. They are family - you can talk about this stuff. Everybody poops. And, if you are reading this, it is likely that it will be YOUR job to unclog the marine head when it gets stopped up.
Now, if you like to use "the good stuff" (like Ultra Soft or Ultra Strong Charmin), put it in the trash. Marine toilets simply aren't as robust as your toilets at home.
In conclusion: rapidly dissolving stuff, flush it (with conservative usage). Anything else, bag it. If you think a small garbage bag of used toilet paper is gross, you have never had to deal with a stopped up marine head. (One job I had, the head got stopped up twice the first week on the job. I had to take it apart and clean it, the first mates were too squeamish.) After instituting the "nothing but pee or poop in the toilet" rule, no more clogs. For years.
Yeah, I know: that's enough of that crap.