This question is about the effect of available information on random quantum channels.
Suppose there are two black box devices.
Device 1. We have a black box device with a single qubit in it. Once we turn it on, this device does the following noisy quantum channel on the qubit $\rho \to (1-p) I \rho I^\dagger + p X \rho X^\dagger$, with certainty. Here $p \in (0,1)$. No measurement takes place after this device.
Device 2. This black box device has an identically prepared qubit in it. Furthermore, within this device there is a physical switch hidden that can be in state 'Heads' or 'Tails'. If the switch is in state Heads, nothing is guaranteed to happen; and if it is in state Tails, then the quantum bit-flip channel is guaranteed to happen. The switch is operated internally according to the outcome of a classical probability distribution, for example by a biased coin that flips Tails say with probability $q \in (0,1)$. Once we turn this device on, the device flips the coin internally, and therefore thus executes either $\rho \to \rho$ with probability $q$ or $\rho \to X \rho X^\dagger$ with probability $1-q$. No measurement takes place after this device.
We haven't yet but are about to turn on both devices.
Question 1. Suppose we will be informed of the outcome of the coin flip. If $p = q$, in this moment in time (so prior to turning on either device), are the two processes that are about to happen different or identical from a modeling perspective? Critically, assume that we do have access to all of the information above.
Question 2. Suppose we will not be informed of the outcome of the coin flip. What would be the answer now?