is it true that predatory birds do not hunt near their nest sight?

Answer 1

This statement seems to agree with that:

"Raptors may also permit small birds such as House Sparrows and kingbirds to build their tiny nests in the raptors' ample platforms or to be close neighbors. These small birds sound the alarm when an intruder approaches, but are not threatened by their predatory "hosts," which ordinarily do not hunt near their own nests."

As does this:

"Woods where Cooper's hawks nest may remain heavily populated with songbirds, as these hawks hunt away from their nest area."