There are no real evidence for the Soviets sending unamed men or formations (like in the movie "Enemy at the Gates") where half were armed and half were not into battle in ww2. While absence of evidence is not evidence of absence, and there might be a few isolated cases, in general, the Soviets never had problems supplying their troops with arms. The myth could have a few (or all of them) sources. In ww1, the Imperial Russian army suffered a severe lack of arms - all countries in the conflict did, as their respective industries could not re-tool quickly enough to supply the armies for the much longer than expected total war. The French drilled their new conscripts with wooden mockups and at times delayed the deployment of new forces (but armed them before sending them to the front). The British had to purchase Canadian and US rifles and took rifles from the Indian army as a stop-gap measure to make up the shortage back home, the Germans modernised older Gewehr 88 rifles and the Russians bought tens of thousands of Japanese Arisaka rifles. However, unlike the other countries, the Russians, being mostly isolated from the rest of the world due to the Germans controlling the southern Baltic Sea and the Ottomans the Straits of Bosphoprus (the trans-siberian railway could not take heavy traffic and was single track and the port of Romanov-na-Murman/Murmansk was not connected until 1916) had large problems supplying their troops with rifles. Despite the purchases and deliveries of British and Japanese rifles were made (remaining rifles in 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.

Ammunition was also an issue, as bottlenecks in production and for the Central Powers and Russia no access to the world market of nitrates (mostly from India and the guano caves in South America), meant that gunpowder was hard to come by. The Central Powers solved this by industrialising the Haber-Bosch process to produce nitrates from air, while Russia got deliveries from their allies.

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 late 1941 and early 1942, 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, which coupled with Red Army losing most of their pre-war force and all of its equipment with it, making it necessary to raise and equip new troops. During this time, Soviet troops often experienced going into battle equipped with far less heavy arms than their official table of equipment indicated - but the shortage would be in artillery, anti-tank guns, AA guns, mortars and perhaps machine guns rather than sub-machineguns or rifles.

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 and were well equipped.

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 nor Narodnoe Opolcheniye.

Another possible source of this myth is the Soviet "human waves". Early during the German invasion, the Soviet doctrine focused on zeal and discipline rather than tactical flexibility and advanced manouvres. Soviet units that found themselves cut off by the rapid German advance would desperately try to break out and massive "human wave" style attacks were part of these desperate actions, and caused extreme casualties, but according to the Soviet high command at the time, it was better that the men died and took at least a few Germans with them rather than surrendering - if they failed to break out, the unit was lost anyway sooner or later due to lack of supplies.

Later in the war, the Soviets developed an advanced operational doctrine, where they would use maskirovka tactics to trick the Germans - extensive nonsene radio chatter, dummy tanks and trucks, single trucks towing dozens of carts made into dummy trucks back and forth, creating roads and dust clouds would trick the advanced German radio interception and air recoinnasance and cause the Germans to reinforce the wrong areas, while overwhelming forces were moved to another area, carefully hidden and then hurled into the fray. The Soviets suffered massive casualties during the war, and had a lower number of well-educated from which to draw a cadre of NCOs and lower officers and their troops thus had a lower level of training and less skilled lower leadership and their actions looked like just a massive wave of men and weapons to the Germans, who never saw the advanced build up and the flexible switching of reserves and heavy assets belonging to high command to successful attacks and the suspension of less successful ones that casued the Germans to be "drowned" in these waves. German veterans in West Germany were available to western historians, and many German officers and generals published their memoirs after the war in the west - while the Soviet side of the story generally remained on the other side of the Iron Curtain until the 1990s, cementing the German ignorance of the operational doctrine behind their defeat in the west.

So, the bottom line is that in ww1, the Russians did send unarmed byt uniformed troops into combat. In ww2 it might have happened for Narodnoe Opolcheniye home guard units at times when the Germans over-ran their home regions, especially late 1941 and early 1942, but there's no real evidence of the Soviets conciously sending unarmed men into combat. The idea can have been reinforced by the Germans seeing desperate massive human waves attacks by the Soviets early in Barbarossa and confusing them with the more advanced operational assaults later in the war, since they could not see the preparations and work behind them. The German idea would then be reinforced in the west since the German experience got out to western historians while the Soviet perspective did not.

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.

There's an older answer on this question here, but essentially the whole 'one rifle for every second man' is a myth. The Soviets, for the most part, were quite capable of supplying weapons for the soldiers, and the only instances where the trope of unarmed soldiers came into play was in the opening months of the German invasion. During Barbarossa the Soviets were completely unprepared for an invasion and undergoing significant reform in the Red Army. As such, the rapid German invasion often surrounded vast pockets of Soviet soldiers, some of which had not received enough weapons for all their soldiers yet. So while Germans attacking underequipped Soviet units did happen, it was an exception rather than the norm, and only occurred at the start of the war when the Red Army was completely unprepared.

It's also worth noting that having all the weapons in the world doesn't help if your logistics are not up to snuff. Those sixty million Mosin Nagants sitting in factories east of Moscow aren't any good if you can't move them quickly to soldiers hundreds of miles west.