Interesting question! There's a few ways we can go about looking at this.
We can look at the total number of deserters relative to the total manning of the army across the duration of the war. In the case of the British forces, this is fairly easy to find. They list 13,301 deserters and absentees but it's harder to determine the total number of soldiers because some records were lost. Total manning is 6-7 million. That means that the rate of desertion in the British army was around 0.2%. Interestingly, 276 of those British were executed and 30 others from Canada & New Zealand; no soldiers from Australia were executed although some 466 were court martialled for being AWOL or for deserting and 121 of those had a death sentence applied but commuted (this was due to the Australian government's defense act which prohibited execution without permission of its governor-general - such permission was never given).
It's much harder to find desertion figures for other countries. France executed 600 soldiers, I cannot locate how many more actually deserted. The same source suggests that 150,000 German soldiers deserted but logic compels us to think that most of those would have occurred in the final stages of the war when German defeat was inevitable.
So, overall, the number of desertions was not high compared to the army sizes (Britain 6-7m, France ~8.3m, Germany ~11m) and there were many barriers to desertion. Those barriers would be:
To summarise then, overall desertion numbers were very low compared to the size of the armies. Success was quite high if you count "success" as deserting without being executed.
However, maybe there is another way of looking at things. There were many mutinies during WW1, the most famous of which occurred in French forces in 1917. You can find a concise list of other mutinies here (not all on the Western front). In addition, Australia had several and the British also came in for their fair share (see here and Etaples.
Germany famously had a naval mutiny (the Kiel mutiny) as well as other incidents as the war drew to a close.
The linking factor of most of the mutinies was that they generally occurred after the beginning of 1917. It was at that time that war weariness was increasing, supplies were short and a general feeling of hopelessness had set in across the Western front. The common soldier felt that his life was being spent to no purpose or, worse yet, to build the reputation of a general who had never seen the front. Famously Petain quieted the French mutiny in part by vowing that no further suicidal attacks against prepared positions would be undertaken and that soldiers' lives would be spent carefully.
In that context, soldiers were widely dissatisfied with their place on the front, their leadership and the conditions that they lived and fought under. Perhaps the only thing that kept mass desertions from happening was the sense of camaraderie that developed when standing literally shoulder to shoulder with others for months in appalling conditions.