I hate to break it to you but a detached house doesn't mean a detached (as in a gap between you) garden!
We live in a detached house. Our garden is just separated by a fence from both our neighbours (also both detached houses). That's the norm for detached houses - you don't have land in between!
Which means we can hear them whenever we're in the garden or have the windows open. We have a retired couple one side and a young family on the other. The side that has the retired couple just has a wire netting fence for the lower part of the garden as we have open countryside behind us so it allows us to all enjoy those views across each others gardens.
But we have a privacy fence (6 foot high) near the patio which has always been there and they've always had their shed at the end of that. So you'd physically have to go to the middle of the garden and lean in to see either patio. The younger family bought off of dog owners so they have high fences.
You could do something like putting in a shed or similar to block their view of your seating area if that's the issue. Or, as others have said, just say something like you had clematis etc in your last garden so you're installing higher fences to grow some up without it coming over into their garden. You can't grow clematis or honeysuckle on low fences. Plus it'll look and smell lovely!
But the point is, we can still hear hear our neighbours if they are in their garden and they can hear us. Over Easter we had family over and enjoyed sitting on the patio chatting. We heard the younger family's kids squealing and giggling and the murmur of voices one side - it was lovely to hear the kids having fun. Our retired neighbours had friends over and we heard plates clattering so we assume they enjoyed lunch on their patio too. It was Easter, it was sunny and we all had time off so why on earth would we ask permission to use our own gardens?
As to this thing about hearing your conversations with your DH - are you using a megaphone? We can only catch murmurs and laughter. I have no idea what they're actually saying.
You don't have to be super friendly with your neighbours. But having a chat is useful, we all know when the neighbours are away and will keep an eye on the house for example. Just be polite and for goodness sake enjoy your garden!