The answer to the former question is rarely. Considering the number of troops fielded by the German Army, roughly 5,000 of the well known MP18 saw service during the war though approximately 10,000 were made. These were mostly supplied to German Stosstruppen, or stormtroopers. Considering the low number manufactured, very few of the millions of soldiers that served in the German army would have even seen an MP18 let alone used one.

More common were the Light Machine Guns used by both the Allies and the Central Powers such as the Lewis Gun,. These were far more common but still fewer soldiers would have used these then the standard issue bolt actions rifles supplied to most soldiers on all sides.

My answer only covers the French army since it is the only one I thoroughly studied.

How often would a common soldier have an automatic weapon ? Short answer : it depends (greatly) on the year of the conflict. In 1918, the French infantry equipped with light-machine guns attacks in combination with light Renault FT tanks, covered by the fire of machineguns sections, mortars and light 37 mm artillery. Planes are flying overhead, designing artillery targets and harassing the enemy. Armored cars are ready to pursue the enemy along with bomber-planes.

In 1914, soldiers in red pants are charging with bayonets, with a set-distance of one pace between each other, very much like Napoleonic soldiers.

What happened in between?

In 1914, the first battles are catastrophic for the French army. On the 22th of August alone, 27 000 French soldiers are killed. The over-reliance on the fantasized élan of French soldiers showed its limits. Everyone, from soldiers to HQs, rapidly agrees that this war won’t be won because your soldiers have, pardon my French (sic), the biggest balls, but through a scientific approach of war. And that firepower is now king over melee : the pre-war conception was that soldiers with bolt-action rifles+bayonets were pikemen with a 1000m pike. In a month, this idea is over and done.

As soon as 1915, the first fusil-mitrailleurs/ light machine gun (Chauchat) appear among units, before being standard in 1916. From 1916 to 1918, a french section (40 men) has six light-machine guns, and some of the soldiers have access to semi-automatic rifles too. The difference in firepower from 1914 is tremendous. Regarding machine guns : in 1914, you have 6 machine guns per regiment. In 1915, 8 machine guns. In 1916, 24 machine guns.

The firepower gestion is also heavily decentralized during the war. In 1914, machine guns are treated almost like artillery at the regimental level. In 1918, a French section can be supported directly by them. This allows for more flexible tactics then the heavy 1914 formations. This goes outside of your question, but consider also flamethrowers, mortars, rifle-grenades, light-artillery added to the mix. Automatic weapons are just a part in an tremendous increase in the firepower output of the French infantry.

Increase in firepower was also a solution to a decrease in available soldiers : from the end of 1915, the French army loses 4 regiments a month. Men are a rare resource, but the industrial machine is now roaring : its preferable to spend equipment and materials than men. This is an incentive to better equip your soldiers.

In conclusion, if most soldiers didn't use an automatic weapon during WW1, the chance to use one, inside a section, went from 0 in 1914 to 1 in 6 or so at the end of the war.

*Sources: * Histoire militaire de la France T. III De 1871 à 1940 sous la direction de Guy Pedroncini, Presses universitaires de France, Novembre 1992

La chair et l’acier, L'armée française et l’invention de la guerre moderne (1914-1918) Michel Goya - Tallandier Editions, 2004