You could have another adjective, e.g. inga egna rödhåriga barn 'no redhead children of their own' but not a possessive pronoun.
It's just like in English I have no books OR I have my books OR I have no good books but not I have no my books.
This no (the no as in no book as opposed to yes or no) doesn't go together with possessive pronouns because that would cause a clash in definiteness. Just like you can't say I have no the book so you have to say I do not have the book instead. One can say that this no is sort of short for 'not a'. But possessive pronouns can only be used about things that are known to some degree – when you know an object well enough to know whose it is, it isn't just "an object" anymore. This works the same in English and Swedish.
I could be wrong, but I think the difference between sina egna and sina is like in english where we have "They have their own interpreter" vs "They have their interpreter" respectively. Assuming we're talking about the same they in both instances, the first sentence is pretty straightforward, They have an interpreter of their own, while the second sentence could mean They have their interpreter (with them), but it could also mean They have an interpreter (who they can call their own).
(Don't quote me on this. Someone correct me if I'm wrong, please ^.^)