The manager is responsible for the following:
The designation of persons working under his or her supervision.
Their own qualifications for working with the type of firearms being used.
In this context, does “their qualification” mean the qualification of the persons designated by the manager? Or does it mean the qualification of the manager?
This is what we call an ambiguous antecedent. It has a relative pronoun, their, but the text before it gives two possible referents. It could mean the manager, or the “persons working under his or her supervision”.
Often, you have a syntactically ambiguous antecedent, but it’s not a problem because the semantics make it obvious what is referred to. In this case, however, it is well and truly ambiguous.
One clue we have as to what they mean, however, is the use of his or her in the previous bullet point. That indicates that the author doesn’t use the singular they, meaning that their in the next point is most likely plural, and therefore must refer to the “persons working under … supervision”.
On the other hand, we also have their own, which suggests some sense of it being reflexive, that someone is responsible for something of their own. That would suggest it is the qualifications of the manager.
So, it could semantically be either, it could syntactically be either, and the minor pointers we get in the text are pointing in different directions.
If it’s a legal document, or a policy, or whatever, we can finally fall back on the idea that items in a list should stand independently. You should be able to read each of them along with the text before the list and get meaning from it without needing to read the others, unless explicit reference is made. That would suggest it’s the manager’s qualifications. However, exactly what counts as an explicit reference varies between authorities, so their could be considered such a reference.
Basically, whoever drafted that needs a stern talking to. If I had to decide what I thought it meant, the reflexive own sways me and I think it means the manager’s qualifications. But it should be written less ambiguously.