*Chestnut, our next door neighbour has a huge forest sized tree about 40/50 feet from their house, we get the leaves, but are under no threat from the tree. They had tree surgeons in about 5 years ago who drastically lopped and cut back most of the head. It is now growing back again and may need to have the process repeated in a few years.
To be fair, as the prevailing winds are westerlys, if it had come down in any of the recent gales, it would have fallen away from the house.
I think the first thing to do is check whether your house is to the east of the trees, which would give you some guide to the danger, but otherwise both need to be pruned to reduce the danger. i i was the owner of the conifer I would have it out, for the simple reason that they block so much light to windws in the shade of it and, in this case, is not very pretty. A tall shrub would provide as much privacy and less danger.
I do think the real problem is climate change which has meant that over recent years that these winter storms with exceptional winds are becoming more frequent and stronger.
We are responsible for fences on all three sides of our garden. When we bough the house, the fence was made up of interwoven panels and, mostly, wooden posts and we had no problem at all with it for probably 15 years, the occasional panel, but not much. Gradually, since then we are having more and more panels blown out by the wind, we replace them and they blow out again.
For the last 5 years we have, slowly, bit by bit, because it is expensive, been replacing all our fences with vertical boarding, concrete posts and concrete gravel boards and these are standing up well to the worsening weather, but, my goodness, it is expensive and by the time the whole garden is done, a few years from now, we will probably have spent £10,000 on it.