In 1941 the British troops retreating from Greece to Crete were forced to leave their heavy artillery behind. In an attempt to prevent the Germans from making use of those guns, the British artillerymen took the gun sights with them to Crete. When the Germans invaded Crete those artillerymen were left with sights for non-existant guns

In the famous Doolittle Raid, the tailguns from the US bombers were removed (in order to keep weight down) and replaced with broomsticks painted black.

Despite both of those, your history teacher is unlikely to tell you the British sent their men into battle with artillery sights but no guns, or that the US sent bombers on missions without giving them weapons to defend themselves.

In war, there can be times when soldiers have to fight with less than ideal equipment. Just like the British artillerymen, there were times when Soviet infantry were forced - due to circumstances on the ground - to fight without sufficient arms or ammo. But it wasn't standard practice.

The point is that it's easy (and unfortunately common) to point to examples where the Soviets were forced to fight (or in this case train) without sufficient equipment and act as though that was the norm, while almost certainly not doing the same for the Western Allies.

So I'm not saying your history teacher was right or wrong (I don't know a reliable source that would confirm their story, and don't know how you could show it never happened) but it's important to remember that context is always important.

I answered a similar question here, and this reply is a slightly modified version of it.

First of all, training and doing drill with wooden guns or broomsticks (or other sticks) have been common in many armies throughout history. Frederick the Great's Prussian soldiers drilled with sticks before being issued muskets. Parts of the US army conscripted in 1917 initially drilled with wooden mock-ups of Springfield M1903 rifles and so on. When learning how to handle a rifle and drilling to commit movements with the weapon to your muscle memory, it does not matter that much if it is a stick or a gun - and guns were often expensive or in shortage.

Did troops train with sticks? Did they go to battle unarmed?

It is true for the Russian Imperial Army during ww1 and some for the Narodnoe Opolcheniye (people's militia) units of the Soviet army during ww2.

The Russian Imperial Army was unable to equip all its drafted men with rifles early 1915 - and purchases and deliveries of British and Japanese rifles were made (remaining British and Japanese rifles in Soviet stock were later sold at high prices to the desperate Spanish Republic 1936-38). By December 1914, the Russian Imperial Army had roughly 6,5 million men in uniform, but only 4,5 million rifles to equip them with. At the end of 1915, they had 2 million men at the front, but only about 1,2 million rifles to equip them with. However, by that time the Russian production had picked up, reaching about 1,3 million rifles per year, and there were 2 million rifles in Russian ports or on their way from Britain and Japan.

All countries fighting in ww1 suffered huge problems trying to supply their forces and equip men raised to replace the extreme casualties suffered by the pre-war armies during the first year of the war. It took time for industry to change to war production, and the lack of nitrates (in the Central Powers due to the blockade, in the Entente due to difficulties in increasing shipment of guano) hampered production of ammunition. The British had to purchase Canadian and US rifles and took rifles from the Indian army as a stop-gap measure. However, the Russian Imperial Army was the only army to send uniformed but unarmed men into combat, expecting them to arm themselves on the battlefield late 1914 to late 1915.

During ww2, the Soviets attempted several times to use Narodnoe Opolcheniye, or people's militia units as a desperate stopgap measure when the Germans advanced deep into their territory. These were civilians organised in a sort of a home guard and at times trained decently well and equipped almost like regular troops and at other times thrown into combat barely equipped at all when there was little or no time for training and little equipment available. This was especially problematic during Autumn 1941, when much of the Soviet war industry was being moved from territory that had been or would be over-run to beyond the Ural mountains, causing a temporary slump in supply and new equipment available to the Soviets. For example, the German 6. Panzer-division, when advancing on Stalingrad as part of Fall Blau in summer 1942, encountered a Soviet unit of AA guns and attacked it, noting that the enemy was firing badly, and after running over the unit, discovered that the guns had been manned by female factory workers still in civilian clothing. On the other hand, the Narodnoe Opolcheniye units part of the 62. and 64. Army that defended Stalingrad seem to have performed well during the fighting.

It is quite possible that Narodnoe Opolcheniye units were at times equipped with a lower number of rifles than it had men and went into battle like that - they often lacked heavier support arms such as artillery, AT guns and mortars. However, I have not seen any records of systematic under-equipment of Soviet units during ww2, neither regular forces or Narodnoe Opolcheniye.

The bottom line is that many armies throughout the history of gunpowder warfare have used wooden mockups, broomsticks or sticks to drill troops before issuing them weapons, including the US army in 1917. However, the Imperial Russian Army is, as far as I know, the only regular army to send trained and uniformed, but unarmed regular units into battle, expecting them to arm themselves on the battlefield.

However

Sources:

Brusilov Offensive, by Timothy Dowling.

Race to the Front, by Kevin D Stubbs.

Stalingrad, by Anthony Beevor.

2194 days of war, by Cesare Salmaggi and Alfredo Pallavisini.

https://www.army.mil/article/185229/world_war_i_building_the_american_military