Did they fall on purpose
Yea I believe they fell on purpose. However I do not believe that somehow Sae had some influence on Mayu to make her jump as a kid(as some people who will not be named have suggested)
The thing is kids do stuff without thinking through, if you try to denies this remember you were, or in some cases still are, a kid and also acted without thinking. Mayu didn't want to be left behind so she went on a impulse, without thinking, and 'slipped'. While I don't think breaking her leg was intentional the fall was. As it would make Mio come back for her, which she did. Also we see her in the Frozen Butterfly ending she clearly happy she hurt her leg, which she also expressed during the fight with Sae.
Same goes for Sae as she wanted to go through with the ritual and stay in the village. So she 'fell' knowing it would be hard to get up, or if anything slow them down, so the villages who they saw looking for them would find them. Her's was a little more thought out, just a bit, and judging by the fact she didn't hurt herself while falling she had a bit more control over it unlike Mayu who did it in the moment.. Please keep in mind I said 'a bit' and I can't say she had control or not. You can right yourself as you fall but it not easy and this is speaking from experience.
Granted said experience was falling many times after jumping from boat to boat but same difference.
Also falling, even on purpose, is scary. You just know 'I'm gonna fall” you don't know the outcome till you land. Even if you have control there a bit of doubt. It that unknown outcome. Even when Mayu work up she thought 'I'm alive” because she wasn't sure what would happen. So don't assume that because they scream they weren't scared and it wasn't on purpose. Also a scream would mean Mio/Yae would...or should..hear them.
Nether wanted to be away from their sister. Both characters will go to the extreme to get what they want.