No, there really isn't much. During the Wronging phase, everyone is thinking about what cards to draft AND what gangs others are playing on at the same time. During Montage phases first you dice draft and there s room for negotiating which results get taken by who etc and figuring out which results will be taken by others and which to take immediately etc. When it comes to the Montage card playing again, you need to figure out who's going for upgrades if you want to get in first and how much speed you want to bid on (there are worse Montage card values for better speed of each type). When you come to resolve the Montage everything apart from Upgrade buys can be done simultaneously so again, zero downtime.
During the Fights you need the player adjacent to you to check minion and boss activation, so they're busy, but we found during tests that even though it's a competitive game everyone ends up getting involved trying to solve the puzzle. It was counter-intuitive to our designer - why are all the test teams helping each other complete the den when they're in competition? And the answer is obviously: it gives some players more satisfaction to solve others' fight puzzles than keeping the competition at bay. Whatever the case the fights are short.
Hardcore rules (which have not yet been released) which might unlock as a Stretch Goal also include a timer for Fight rounds which makes the action frantic.
Finally, with regards to player interaction, it might not be obvious in the first play through, but to play this competitively you need to have an idea of what the other players are going for. Sometimes going for a lower level boss match or matching the gang but not the boss card (and thus potentially scoring only den points) is worth a lot more than the obvious higher VP option you have if that means denying one or more of your opponents high VP yield on a boss they're gunning for. Figuring out what they're gunning for comes out of the drafting, their recon and fight choices and Vengeance cards in play.
In summary - downtime is low and indirect interaction is there once you get familiar enough with the game to figure out what others are doing.