At least which schemes like Reed-Solomon, the "ECC bits" conceptually are not a separate piece of data that is computed and then added to the data to be protected, despite that being how it is usually implemented. The algorithm really "expands" the data, so the redundancy is spread out across all the bits. Thus any bit errors anywhere in the whole encoded "codeword" can be corrected, up to the limits of the code.
I believe they use Reed-Solomon encoding but as it is up to the memory controller to implement it can change. The memory just provides the extra bits for the software to use. In our fictional gun ownership example it would be like the owner hiding multiple guns around the house so that if one was compromised by an intruder they could still defend themselves. :-)