It is not a legal requirement give the details of insurers. I can see why too. If there is some dispute and you believe it is not your fault then to stop the inurers doing some sort of a deal between them, which they will do, then withhold the details.
I imagine she thinks the accident is your fault and does not want to her insureres to get involed in the argument and do some sort of deal, the costs of which always end up costing the punter.
I am curious as to why you are claiming from your own insurers, are you accepting blame?
Personally I have had to claim off other motorists, I don't even tell my insurers, they do not need to know unless you are making a claim. They always say you should tell them but that is only their own internal admin rules, not a legal requirement.