How exactly do umbrella insurance policies work?

[Jeff L.]

Say I have two properties, both under separate LLCs and insurance policies.

If I get an umbrella policy under one of those insurance policies, can it cover the other property as well? Or do I need to get two umbrella policies, one for each?

If I get an umbrella policy personally (like under my home residence policy), can I have it cover my LLC properties? Conversely, if I get it under one of my LLCs, can it cover me personally?