It seems your trees (yes, Leyland cypresses are trees) were never properly trimmed, more likely planted once and then left alone.
Which means they behaved like all trees planted closely together: Gained height quickly, losing density. And all conifers tend to become bare on the bottom and inner branches.
Now, the "almost fake" looking hedge you are talking about is the result of the exact opposite gardening approach: It requires constant trimming of the outer branches (a bit like just nipping off the tips) to encourage more and more sideward growth which will result in that super dense green wall.
Typically the first trimming should happen in spring, just before or right when the trees start to grow. A second trimming is optional, if you choose to do so, late August or early September would be a good time to neaten up your hedge. Always choose a frost-free period and, especially for the second trimming, ideally an overcast day to minimize stress, both for the tree and the gardener.
Unfortunately, your options are limited: Unlike most deciduous trees, the cypresses won't regrow if you cut back to the bare branches. If you want to keep your trees, you can start to trim the sides, but as I said, do not cut into the bare wood. The hedge will become a bit denser, but especially the older trees will never give you full coverage again.
If privacy is really an issue, you might consider biting the bullet and replace the old overgrown tree with new ones. Leylandii grows so fast, even with regular trimming you should have the desired green wall in just a few years. Especially if you buy larger (=pricier, unfortunately) new plants.
And if you are looking into replanting, you could also consider other hedge plants - just an encouragement to take a unsatisfactory situation and turn it into an opportunity.