They often don't know how to react to compliments.
or
They don't often know how to react to compliments.
Which sentence is correct?
They often don't know how to react to compliments.
or
They don't often know how to react to compliments.
Which sentence is correct?
It's often the case that often implies more than an insignificant number of occasions, but less often than the much stronger usually (which invariably means more than half the time).
Since by default, often modifies the immediately-following element, this leads to a difference in meaning between OP's two examples. Taking the same construction and applying it to a simplified statement you just made about your wireless printer, for example..
1: It often doesn't work
2: It doesn't often work
Now let's suppose that after I heard you make one of those statements, I was invited to place a bet at 50-50 odds on whether the printer will work the next time it's used. You can be quite certain I'd bet on Will work after #1, but I'd bet on Won't work after #2. Either way, if I didn't win my bet, I might well accuse you of deliberately misusing language to mislead me into making a bad bet.
Any native speaker Many native speakers1 would interpret #1 as meaning that although there are some (annoyingly frequent) occasions when it doesn't work, usually (more often than not) it does work.
On the other hand, #2 would be interpreted as meaning that more often than not it doesn't work.
I have to admit that distinction isn't so clear-cut for me with...
3: It doesn't work often
...which I'd normally interpret as being closer to #2 than #1 above. But because it's inherently less idiomatic, I might well struggle to understand exactly what was intended.
1 Apparently not all native speakers think the same as me, as reflected by this comment below. I will admit I'm often wrong about such things, particularly if there's a US/UK usage split I'm unaware of. But linguistically speaking, I think that admission is perfectly consistent with my ongoing claim that I'm more often right than wrong. Whatever - let's just see how the votes go.
To spell it out for OP's exact examples, I think the first version implies that usually they know how to react, but sometimes (a not insignificant number of times) they don't. Whereas the second version implies that it's actually quite unusual for them to know (but sometimes they do).
The main point, I think, is not the appropriateness of "often" but its correct placement in the sentence. The correct placement is "They often don't know how to react to compliments." "Often" should not be placed in the middle of the verb phrase ("don't know").