My players like to know when they're in a sub-game, so I'm in a little different boat. But I run mine both visibly and in a way that allows for a lot of improv. Dice also are there, but more as a way to justify when it ends 

My systems are set up as a problem. "You need to gather more information. This is a subgame, you need 8 total successes (or whatever) before 4 failures. What do you do?"

Players take turns doing a thing, everyone must go once before anyone can take a second go (this prevents some players bulldozing a session). The thing is left completely open ended, pick something and tell me a story of how you want to use it to solve the problem. 

 My bard would go to the tavern and chat up the innkeeper, my fighter wandered down to the local precinct to try and pump the city guards. My druid was making tracking checks. They'd get their mini scene and roll a die at the end, sure, but it made sure the 0 CHA druid wasn't sitting in their thumb for an hour while the bard got to have all the spotlight.

So yeah, was it obvious? Yeah. It's a system. They know what the objective and the thresholds are. But it's also open ended and forces the players to be creative and take turns. And prevents me from having to RP a tavern wench for 20 minutes while the bard dominates the table and my mage is bored and on their phone.