I've used both methods. I've done some hatches on their sides and i've even split a hatch with some on their sides and some in cartons. I haven't noticed any significant difference in hatch rates on the different methods however I have noticed a few things. First off, if I am hatching shipped eggs, I tend to use cartons b/c it seems like it works better for the ones that half misshapen air cells. I have noted that it tends to take longer for them to zip around and pop the top off but once they do the chicks are stronger and start perking up quicker.
If I am using my egg turner I often use cartons too b/c then the eggs stay in a position they have been throughout incubation. If I hand turn I often leave them on their sides to hatch to keep that same condition from incubation.
Using the egg cartons really does make a HUGE difference in clean up. Another thing that helps with clean up is using some of that non sticky shelf liner (it's blue in the pics below). I find that liner helps the chicks get a better footing and makes clean up a lot easier, regardless of which method you use. For $5 I can get about 10 cycles worth of eggs so I don't feel guilty tossing it out after hatch. Only thing is to make sure there is room around the edges to still be able to add water in case I need to up humidity during hatch.
If I put them in an egg carton I use a cardboard one and use a pair of scissors to poke holes in the bottom of each cup for ventilation.
Another thing I have been doing the last few hatches is to trace around the air cell before lockdown so I can see where they 'should' be pipping. Interestingly enough, it is often a great predictor of where they will pip and the zip line around the shell, almost like I'm tracing a path for them!
here are some pics of incubations in cartons, sorry for attaching so many, found it hard to just pick a couple! I'm a picture nut so I have a ton.
good luck with your hatch and feel free to PM me if you have any questions!