Nazi Germany during the war set out to kill as many Soviet prisoners of war as they possibly could, the message broadcast in your question would have been pure propaganda. More than half of all Soviet prisoners died in German prison camps while other Allied prisoners in the camps had a death toll of less than one man out of every twenty, according to Prelude to Nuremberg: Allied War Crimes Policy and the Question of Punishment.

The Germans also killed an estimated two million Soviet POWs at capture. Again far outstripping any estimates for the POWs of other Allied belligerents. The impressive death toll for recent Soviet POWs during the war was greatly influenced by the Kommissarbefehl, the Commissar Order, which was an order issued by Hitler that detailed what was to happened to Soviet political commissars when they were captured, but crucially, it also applied to any prisoners that could be identified as being "thoroughly bolshevized" or representative of Communism. The average, highly politicized, common German soldier would interpret these directives extremely widely, often with support from junior officers. In fact, the orders explicitly stated that German soldiers who failed to carry out the breaking of Soviet prisoners with force without using their weapons or without sufficient energy would themselves be punished.

This order was further reinforced with the Aussonderungsaktionen or the "Weeding" policy of 1941-42, where the Gestapo would investigate prisoners of war and identify people who were racially, academically, and crucially for the Soviets, politically dangerous or undesirable. Those so designated were executed, often summarily, but sometimes on a far larger scale by the SS. A large number of Soviet POWs were also immediately transferred to concentration camps. Soviet citizens feature prominently as some of the earliest victims of gassing chambers, and Zyklon-B.

At the end of the war, a considerable effort would be made to claim that the German army never enforced the Kommisarbefehl or the Aussonderungsaktionen, that these actions were inherent to Nazi ideology and that the S.S. or the Gestapo were to blame for the consequences of both orders. These claims were entirely dismissed but did give us the start of the "following orders" defence at Nuremberg. Which was also ultimately dismissed. There were also vocal High Command opponents of the measures, who claimed that without the orders, the Soviets would indeed be easier to defeat, as the average Soviet soldier wouldn't be placed in a fight-or-die situation.

Another key in understanding why the Germans were so brutal with Soviet POWs lies in the politics behind the war. By rote, Germans, particularly with the common soldier, believed that the victory over the Soviet Union would be quick. Hitler had been clear, the Soviet Union was populated by beasts, lesser men, that could not hope to match Aryan soldiers in the field, and their political ideology provided them with a corrupt, inhumane, easily toppled national framework.

Not until 1942 would Hitler see better treatment for the Soviet POWs as an option, and only then it would be because his ideas about the Soviet Union outlined above would prove to have been rather wrong. The Germans would get stuck in a far longer conflict than they had envisioned and a need for more workers would emerge.

So, the urge to ship the prisoners to Germany or even keep them alive at all, was considerably lessened. Why should the army burden itself with cattle that would have to be thinned after the short war anyways? Why even bother sending the prisoners back to Germany when all of this land would soon be German? As the war progressed, the Soviet POWs became more and more like chattel to be used in the habits of the German High Command. In brutal cases of rigueur oblige, in war, you have to use everything you can manage to get your hands on. So, the meager winter supplies of the average soldier started being supplemented by captured Soviet gear. Which further exacerbated the death toll for Soviet prisoners of war since Russenlagers, Russian camps (contrasting with Oflags, Marlags, Stalags, there were special camps for Russians, not just separate enclosures in existing camps although those existed as well) in the East were often little more than a barbed-wire enclosure.

The message broadcast in your question to Soviet Prisoners of War, was nothing but false hope, and ultimately nothing but propaganda aimed at shortening the conflict or give the Germans an edge.

Unfortunately, for the Soviet prisoners, their mistreatment would not stop with the war. Those who survived would be viewed by the Soviet authorities as collaborators, or worse, they would be killed as traitors to the Soviet cause. Soviet authorities would investigate and hold all Soviet citizens being repatriated after the war. Fewer than one in five Soviet POWs would go home from what the Soviets called Filtration Camps. Those who weren't sent home would be drafted, sent to the gulag, or executed.

On another note, my grand father was captured by the Germans in 1942 in Dieppe. He spent the rest of the war in different Stalags. He was not undesirable, he escaped the Stalags twice and made his way to France, each time getting recaptured. After the second time he was told there would be no third, so he stayed put.

He wrote in a tiny notepad, that he could hide easily, which he still has, a lot of what he went through, and he showed me some of the things he wrote on his last Stalag which had Soviet POWs in an enclosure that wasn't too far from their own. He described dirty, emaciated men, that they never got any mail, that their food was dumped out of a truck in the middle of place, instead of being served. He has always been of the opinion that the Soviets had it far harder than his kind ever did.

Sources:

The Wehrmacht and the War of Extermination Against the Soviet Union, Jurgen.

The Kommisssarbefehl and Mass Executions of Soviet Russian Prisoners of War, Jacobsen.

Prelude to Nuremberg: Allied War Crimes Policy and the Question of Punishment, Kochavi.

Bloodlands, Snyder.

On June 6th, 1941, German High Command gave the Wehrmacht the so-called "Commissar Order," or "Richtlinien für die Behandlung politischer Kommissare." Under these guidelines, any captured Soviet officer was to be summarily executed as a means of stamping out communism. The idea was that these individuals were the brightest of the Soviet army and rank, and by executing them it would undercut any Russian initiatives.

Many German officers and generals after the war claimed that the Commissar Order was never enforced, but testimonials from soldiers on the ground say otherwise. Regardless, there was a small camp among the German army who felt that the Commissar order undermined the German offensive, as Soviet soldiers would rarely surrender and instead chose to continue fighting in the face of certain death. However, Adolf Hitler was unmoved by these requests to lessen the order. It was only by May 6th, 1942 that Hitler was convinced by his staff that the Commissar order was helping the morale of the Soviets more than his own troops.

This policy falls under the umbrella of "Lebensraum," which was a concept that said the German people needed to expand to survive. Eastern Europe and Russia was seen as the perfect location for an expanding German people to inhabit, so depopulating the region was undertaken in order to establish room for the new migrants after the war.

Edit for sources:

Ian Kershaw, "Hitler: A Biography", W. W. Norton & Company, 2008.

Richard J. Evans, "The Third Reich At War", Penguin Books, 2010.

Richard J. Evans, "The Third Reich in Power", Penguin Books, 2006.

Claudia Koonz, "The Nazi Conscience", Belknap Press, 2005.

Götz Aly, "Hitler's Beneficiaries: Plunder, Racial War, and the Nazi Welfare State", Picador, 2008.

Where exactly did you hear this?

Edit: Im familiar with the loudspeakers, but with the Russians using them against the germans instead.

I can't be sure about Stalingrad in particular, but the Russians were extremely harsh in their treatment of deserters and surrender. After months of German success in 1941, Stalin wrote his Order no. 270 to address the failing morale of the Red Army. The removal of distinguishing marks, retreat to the rear, and surrendering as a prisoner were all considered malicious desertion by the authorities according to Order no. 270. Commanders were required to shoot deserters on the spot, while encircled troops were made to fight until the death, with surrender being a dishonorable and cowardly act.

Stalin's infamous Order no. 227 also addressed the issue of Soviet desertion and surrender, demanding that the soldiers take not one step back! Under this order, commanders found to have surrendered without an official order were subject to a military tribunal and/or arrest. Order no. 227 also directed the formation of barrier troops, or blocking detachments designed to shoot soldiers who were deserting or fleeing.

The historian Nikolai Tolstoy writes in The Secret Betrayal:

"Hitler himself urged Red Cross inspection of [German] camps [holding Soviet prisoners of war]. But an appeal to Stalin for prisoners' postal services received a reply that clinched the matter: 'There are no Soviet prisoners of war. The Soviet soldier fights on till death. If he chooses to become a prisoner, he is automatically excluded from the Russian community. We are not interested in a postal service only for Germans'."

Then, starting in 1942, the Supreme Soviet established 3-person military tribunals at the front to punish deserters. These tribunals could order death sentences, provided they notify Moscow by telegram, but any other punishment was permitted to be carried out directly. Despite these measures, officers often took things into their own hands. In the book Ivan's War by Catherine Merridale, she writes of many accounts exist of officers abusing their power and overusing the death penalty, including a case where 2 leaderless soldiers were shot by a lieutenant while begging for food. At one point, the commissar Lev Mekhlis was notified of the excessive use of the death penalty on the southwestern front in an underserved army.

If a Soviet soldier was found to have deserted, his family would also have suffered. If the soldier's family was not arrested outright, they would see the loss of pensions or state welfare, and even the withholding of schooling for their children. Many material rights would be taken from families of official 'traitors', which could have included soldiers whose bodies were unable to be recovered.

As mentioned earlier, my primary source was Ivan's War by Catherine Merridale. If you're interested in the Battle of Stalingrad, i'd recommend Stalingrad by Antony Beevor as well.

(This is my first time posting in AskHistorians. If my sourcing is insufficient, please delete if necessary!)

documentaries about Stalingrad there is usually at least one mention of the speeches blaring from Nazi loudspeakers during the battle and how some Soviet soldiers contemplated surrender.

It was mostly the other way around. The soviets had loudspeakers spreading propaganda to the trapped germans. And just like with the soviets, the germans didn't surrender. Why would you surrender to your enemy because of silly nonsense stated by your enemy? Surrendering is usually a last resort. Neither the germans nor the soviets believed the other's propaganda.

Or, in the style of Erwin Rommel, were they actually treated like human beings with a modicum of respect?

The silly myth of rommel aside, nobody was treated like human beings because everyone was starving. Stalingrad was a terrible battle and people ( germans and soviets ) were cannibalizing in order to stay alive. The eastern front was a terrible front and most of the POWs on either side perished. In war, the POWs are the last to get the food, resources, shelter, etc.

One easy way to determine if the propaganda worked is to check the casualty reports and the fierceness of the battle. Considering how fiercely both sides fought and how much casualties were inflicted, it should show you that silly propaganda didn't work in the battlefield.

To expand on the question, making it more broader instead of Stalingrad only:

It is about the Georgian uprising on Texel. These Georgians were stationed on Texel during the war, and the battalion was 1/3 German and the rest captured soldiers from the Soviet Republic of Georgia. These men were given the choice of giong to POW camps or aligning themselves to the Germans. So indeed, some soldiers were recruited after being captured by the Germans and not all were killed.

Source: Dick van Reeuwijk. Opstand der Georgiërs, Sondermeldung Texel

I see answers saying few Red Army troops surrendered willingly and that many Soviet POWs died in the camps - what about Vlassov's army and the Galicia SS? From what I understand there was a sizeable force of former Red Army soldiers who fought on the Axis side later in the war

Can't speak to Stalingrad specifically, but in general Soviet prisoners-of-war in German custody would have been kept in prison camps, which were segregated by nationality (Russians separate from French or Poles) and by rank (enlisted separate from officers). The Soviet government regarded all Soviet POWs as traitors, whether they had actually defected or simply been captured in battle, and as such made no efforts to secure their release, advocate for their good treatment, or facilitate civilian relief efforts by relatives or organizations like the Red Cross. Russian prisoners thus experienced much worse conditions in their camps than other Allied POWs, and many died in camp from starvation, disease, or brutality from the guards.

As the war progressed the Germans formed some special military units made of Soviet citizens (including many POWs), quasi-independent of the German army but still commanded by German officers. Other Russian prisoners were recruited directly into the regular Heer as individuals, or were given basic training as spies or saboteurs and sent back accross the front lines as double agents. Most of those who "volunteered" for such duties did so in a desperate attempt to escape death in the hellish prison camps, and didn't end up actually accomplishing much for the German cause.

But whether they worked for the Germans and then re-surrendered to the Soviet authorities (in battle or voluntarily), or whether they refused, then somehow survived their POW camp and were "liberated" by the advancing Red Army, their fates would most likely have been the same: conviction as a "traitor of the Motherland" and a ten or twenty-five year sentence in the slave-labor camps of the Soviet Gulag. In one of hundreds of prison camps spread across European Russia, Central Asia, and Siberia-- alongside captured German soldiers, Soviet political prisoners, and ordinary criminals-- under conditions scarcely better than in the German camps, and sometimes worse: they were worked to death digging canals, felling timber, building railroads, or mining minerals. Most didn't survive their terms.

Source: The Gulag Archpelago, vols. 1 and 2, Aleksandr Solzhenitsyn 1974