I've seen Mookie cross in front of JBJ to make catches I'd swear we're on the left field side of center. Might just be camera angle though. Mookie is definitely faster than JBJ, so he's probably doing what you're saying. If Mookie doesn't catch them, they drop in for hits. I don't think he's hogging balls out there.
Mookie is definitely "stealing" balls from JBJ. The Sox outfield appears to be intentionally taking advantage of a combination of Mookie's better speed and more accurate arm, along with JBJ's stronger arm as backup. But they do this on the other side, too. JBJ lets Beni take most of the plays in LCF and backs him up, as well.
The reason we notice it more in RCF is that Mookie is a better fielder than Beni, and there's much more real estate in Fenway's RCF for these plays to take place. Also, most of the balls hit to deep LCF carom off the monster to be played by JBJ because the Monster is angled to bounce balls back toward CF. This again plays into the stronger arm of JBJ getting those balls back into the infield.
The main goal is clearly to avoid outfielders colliding when all three have excellent range, and in that it's succeeded. The only unfortunate downside I can see is that JBJ's stats don't reflect what his value would be if he was playing on a team that let the CF 'take charge' in the conventional sense. But the benefits far outweigh that, and JBJ has been a great team player on this, knowing his defensive stats have suffered from it.
But this has been the Sox strategy since at least the start of 2016, and it's working tremendously well. A few games in April of last season had some peculiar plays showing that JBJ hadn't quite fully adapted to being the backup rather than the general of the outfield that CFs usually are. But it has been flawless so far in 2017.