It is well known that the area between the trenches was nearly impossible to cross. That being the case, how would one side start to dig a trench while so close to the enemy?
Here's an earlier answer of mine to a similar question about how soldiers dug trenches without getting shot:
It's also worth adding two points.
First, once trench lines were constructed, they tended to creep closer together to make raiding and launching attacks easier. Less open ground between you and your opponent meant less time being exposed to fire.
Second, defensive positions, even in 1914 when static trench warfare was in its infancy, usually comprised two or more trench lines: a line of firing trenches and a line of cover trenches a little ways to the rear (a few hundred feet or so). When one side attacked, often they'd only end up capturing the first-line firing trenches. The defenders would then fall back and occupy their covering trenches, which then became their first line. The attackers who captured the firing trenches would then, over time, integrate them into their own defensive system (assuming they managed to keep hold of the position).
In this case, opposing trench lines so close together weren't constructed that way. Initially, both firing trenches and cover trenches would have belonged to one side. Then, as the battle wavered back and forth in attack and counter-attack, a piece of trench would end up enemy hands or vice versa.
A good example of this comes from the First Battle of Ypres in 1914. On 20 October, the 16th Brigade in the British Expeditionary Force came under German attack. It had been in that position for about a day and managed to construct a front-line of firing trenches and a reserve line of covering trenches. At 11pm that night, a German attack captured some trenches on the brigade's left. To avoid being flanked, the front-line battalions in the firing trenches withdrew and took up a position in their covering trenches. The Germans subsequently occupied the abandoned firing trenches, making it their new front line. The brigade then spent the 21st and 22nd digging a new line of covering trenches further back.
The key thing to note here is that both sides didn't just collide and decide to dig impervious trench lines right next to each other. The transition from mobile to trench warfare happened over the course of a couple months and involved chaotic episodes of attack and counter-attack as mobility ground to a halt and trench systems became more robust.
Sources: Beckett, Ypres 1914; 16th Brigade War Diary, WO 95/1605; 6th Division War Diary, WO 95/1581.
Edit: My original answer was a bit light on the details, so I added an example case.