Before writing this post I had just finished a Turbine match where I beat a blatant cheater 3-0 (without a pocket medic) before the latter accused me of using ESP. This is one of many experiences I've had doing the same thing over and over again. In addition, I've had my fair share of cheating opponents calling me out for cheating myself. And I thought today I'd share a comprehensive guide on how to do the same and show how you can apply this knowledge your gameplay in general. The next few paragraphs will encompass how I achieved these accomplishments and give you an idea of how you will apply this to your gameplay.

1. Common Sense
We should get this out of the way. Not many people consider the subtle details that your audio, the killfeed, and the match hud at the top of your screen can give you enough information to gague how you should play in the moment. Usually if I'm going for a specific target (say the Sniper cheater I'm dueling) I listen for his shots and see if he's killed anyone on the killfeed at the time of the shot. You do not need ESP for this as the more you play, the more easy it becomes for you to play better. And if you're wondering how I even managed to get the drop on the cheater, I sometimes use my teammates as bait as long as I know where the cheater is before I get the drop on him. Basically in essence you need to use as much information as possible to determine what you can or can't do based on what information is being presented to you. This doesn't just apply to fighting cheaters of course, this applys to playing any FPS in general.
2. Deduction
Of course if this is something you're learning now, you'll struggle. A lot. However the more you practice on it, the easier it becomes to the point where its subconscious to you. Many times when I go for a mid fight on 6s, I can easily tell when we should back out based on what's in front of me just by a simple glance. I always ask myself the following:
i. Do the enemies have positional advantage over us? (high ground for example)
ii. Are we all low on health?
iii. In the current area we are fighting in, does the enemy also have more space covered than us?

Doing this a lot can help your gameplay in many ways and benefit your decision making skills as a whole. You could also generalize this same philosophy with what I have been achieving here as stated in the first paragraph. If the enemy team has more than one Sniper, I glance at the killfeed and listen for the Sniper shot that is closet to me. And if that Sniper shot happens to play at the same time the cheater gets a kill, that shot was obviously played from the same position that cheater is in and you can gague where that player is. Multiple shots on the other hand are harder to determine, but knowing common spots on a specified map and and having gamesense can also help as well.

3. Gamesense
You can't really do much if you're all aim but no brain. Gamesense should not require preemptive thought. Gamesense is purely subconscious and that is achieved by having those same experiences whether it results in a failure or not to the point you know what to do in that specified situation. It is the intuition that is built upon from those successes or failures that give you room for second thoughts and counter calls. And when all of your teammates have that same gamesense, it can save on comms for much fewer calls needing to be made, thus making communication much more clearly.

Conclusion
Not only do these strategies work on cheaters, but they are even more effective against legitimate players. But at the same time these strategies are always used in competitive play because it is considered an efficient way to play. And the notion that "if one person sees you, the entire enemy sees you" goes hand in hand with it.

TL;DR: Use information to determine fights, listen to audio cues, have game sense, use all that information to play to the best of your ability.