How closely are your two trees planted? Very often we see Leyland Cypresses planted too closely together, which leads to competition for water and nutrients. As the trees mature, competition and lack of sunlight to the interior of the tree leads to the decline and browning of the inner branches.
Stressful conditions like drought or cold winters can make Leyland cypresses susceptible to different insect pests and diseases such as seiridium and botryosphaeria canker. http://extension.umd.edu/hgic/plants/seiridium-and-botryosphaeria-canker-leylands-trees
In addition, bagworms are an insect pest of Leyland cypresses. If not controlled, these can also lead to browning and decline. https://extension.umd.edu/hgic/bagworms-trees-and-shrubs
You can prune out brown branches during dry weather. Be aware, however, that when you cut into any Leyland wood without foliage, it will not regrow new foliage. In times of drought (not a problem right now), keep a 2-3" layer of mulch at the base of the trees (avoid touching the trunk) and water regularly to help retain soil moisture during dry periods. These steps to reduce stress will help the trees be more capable of resisting pest and disease problems. Leyland cypresses generally look good for about 10-15 years but they are not long-lived in this area.