I have run into this situation before when fighting in darkness or using the ever-smoking bottle. I love these kinds of encounters since they create such a different situation for the players to interact with. The important thing to remember is that as the DM when a situation arises it's important to be consistent with both the rules and reality.
Kerrick has already covered most of the rules. The missing part is that you can probably guess where someone is from the noise they make, a good example of this is the Invisible condition which outright states that the creature's location can be detected by noise - but how hard it is to detect one person's footsteps during battle is up to the DM.
Kerrick also mentions Opportunity Attacks, other reactions have various different conditions, you will need to check the exact reaction to see if it's possible.
How I manage no-vision encounters
As a DM the biggest challenge is managing PC and NPC positions within the cloud. Some DMs may tell you "if your players are good they won't metagame, just keep the miniatures all plainly visible" but I strongly disagree with that. As soon as someone knows some kind of metagame information, it will influence their decision-making. However players are (usually) willing to accept restrictions without trying to circumvent them. The same is true of the DM, you will see if you read this forum or reddit or wherever, there are tons of DMs who metagame either on purpose or accidentally - in the interest of player agency we should accept the same limitations that we impose on the players.
As for the actual management of the situation, if playing online on a VTT, ideally your monsters would be hidden from the player and the player would be hidden from you. If yours can, problem solved. Mine couldn't so I duplicate the map twice. You look at one map with your monsters in the cloud, the PC in the cloud looks at the other. Neither of you look at each other's maps.
If playing in person, I have the player not look at the map. They can borrow a DM screen to block their view. I pull my monsters out behind my own DM screen and stick them on a grid. Neither of us can see each other, but the rest of the players and monsters can still see each other.
Downsides of these methods
These methods are basically the same. The biggest disadvantage is that it's difficult to model "one way detection" like this. You might have to play slightly fast and loose with positions, if a monster gets hit by a fireball and shouts out in pain, you need to tell the player where the cry came from. It's difficult to describe the exact position quickly, so I give a rough position and tell them to mark that spot. If they say they attack that spot, then even if it doesn't technically line up with the original position of the monster, I say it's correct.
The other big problem is what happens when the PC and monsters end up in the same space. You'd never know since neither of you can see everything. I just accept that. 5ft square is quite big, you can walk past each other without realizing it. I rule that you cannot effectively control your 5ft square, and so it's entirely possible to walk straight past your enemy.
As with all tricky situations, it's important to be consistent. Once you have ruled "you can't pick out the goblin's footsteps in the midst of battle" you shouldn't rule that the goblin can hear the PC. Make sure everyone is playing by the same rules.