Christina Websell writes
But ask/tell your neighbours first, if it's their bush, it's polite.

The prunings also theoretically belong to them - you have to offer them
back (this does *not* mean that you can chuck the whole lot on to their
garden for them to clear up). They might also prefer to do the cutting
back themselves. Also, as you say, it's a privacy screen, and they would
like notice of its removal so that they can prepare for the loss of
privacy.

Goodwill of neighbours is valuable and worth nurturing.
--
Kay