Q:

Is independence subjective? 

I am trying to better understand exchangeability.  Suppose that I would like to run an experiment.  I will pick two people and give each a fair coin and tell them to toss N times in a row.  The difference is that to the person A I will tell that the coin is fair, and tell person B that I don't know whether the coin is fair or not.  But I assure both of them that there is no monkey business going on.  
For person A coin tosses are independent and identically distributed, hence they are exchangeable. 
For person B coin tosses are neither independent nor identically distributed but they are exchangeable. 
What's strange for me at this point is that for the same physical process of tossing fair coins in a row, one person thinks that the tosses are independent and the other doesn't, while I know that they are all IID. 
Does this make independence subjective?  Or is my understanding incorrect? 
Basically B has something to learn about the process while A is unable to learn anything new.  Hence their subjective beliefs affects their assessment of independence. 
Should we assess independence based on an idealized true data generating process or based on what we currently know about the process? 
Here is a similar rhetoric: "Resonance: From Probability To Epistemology And Back"
It seems it depends on whether you are a Bayesian or a Frequentist.  Nevertheless it is quite perplexing that if question was posed as randomly picking a coin from an urn containing one biased and one fair coin (not knowing which one you picked) and then tossing the same coin N times, both the frequentist and the bayesian would probably agree that the tosses are not unconditionally independent, which means they both now assess independence based on what they know about the situation instead of the ground truth that the tosses actually don't affect each other. 

A:

I'm a bit late and good things have already been written, however I didn't see the following mentioned:

For person A coin tosses are independent and identically distributed, hence they are exchangeable. 
For person B coin tosses are neither independent nor identically distributed but they are exchangeable." 

This doesn't depend on the information that A or B have, but rather on whether they view probability as epistemic (i.e., referring to knowledge/uncertainty of an agent, like most Bayesians) or as aleatory (referring to data generating processes out there in the world, like most frequentists).  One can even be pluralist and accept both views as advantageous in different situations. 
In any case, person B can, even with the given information, talk about aleatory probabilities and would then normally model the coin tosses as i.i.d.  with unknown probability.  It is inappropriate to say that "for person B coin tosses are neither independent nor identically distributed", because the epistemic model in which they are exchangeable but not i.i.d.  does not refer to the physical process but rather to the state of knowledge. 

What's strange for me at this point is that for the same physical process of tossing fair coins in a row, one person thinks that the tosses are independent and the other doesn't, while I know that they are all IID. 

But this is not the case.  Person A apparently models what the physical process is, but person B is going on about modelling their own state of knowledge, and does not make statements about the underlying physical process. 

Should we assess independence based on an idealized true data generating process or based on what we currently know about the process? 

From a pluralist point of view that's your choice and there are pros and cons of both.  However once you made your choice, be consistent with it! 

Does this make independence subjective?  Or is my understanding incorrect? 

There are always subjective elements in probability modeling that you cannot get rid of.  This is obvious in subjective Bayes, however also frequentists need to make model decisions that are subjective.  Probability models are always underdetermined by the data.  A frequentist could try to test independence (which doesn't happen very often), but this comes with its own problems and can certainly not detect all conceivable alternative possibilities. 

Nevertheless it is quite perplexing that if question was posed as randomly picking a coin from an urn containing one biased and one fair coin (not knowing which one you picked) and then tossing the same coin N times, both the frequentist and the bayesian would probably agree that the tosses are not unconditionally independent, which means they both now assess independence based on what they know about the situation instead of the ground truth that the tosses actually don't affect each other. 

If you'd phrase this in terms how they model the situation like this and for what reasons, this would seem far less perplexing than writing about what people think the coin tosses really are.  Probability in my view is a tool for modelling the world, not a description of how the world really is.